Pre-Cal 40S April 17, 2009
•
0 likes•436 views
This document discusses logarithmic and exponential functions. It defines logarithms as exponents and notes that logarithmic and exponential functions are inverses of each other. It provides examples of using properties like the product, quotient, and power laws to simplify logarithmic expressions.
Report
Share
Report
Share
Download to read offline
Recommended
Pre-Cal 40S Slides April 11, 2008
This document discusses logarithms and their relationship to exponents. It states that a logarithm is an exponent, with the base of the logarithm corresponding to the base of the exponential expression. It provides examples of converting between logarithmic and exponential forms using the definitions that the logarithm is the exponent and its argument is the power. The document also presents the product, quotient, and power laws for logarithms as methods for simplifying logarithmic expressions.
Automaton
Definition of an Automaton, characteristics of automata,types of automata,deterministic finite automata,non-deterministic automata,Example of Automata,Graphical representation of DFA and NDFA. transition table and transition diagram of DFA and NDFA
Lecture 11 sections 4.3-4.4 logarithmic functions
This document discusses logarithmic functions and their properties. It begins with examples of converting between exponential and logarithmic form. It then covers the natural logarithm function and using transformations of logarithmic functions. The main section discusses properties of logarithms, including the product rule, quotient rule, and power rule for logarithms. It provides examples of writing logarithmic expressions using these rules in both expanded and condensed form.
Theory of Computation FSM Grammar Minimisation and Normal Forms
This was the content discussed for Theory of Computation on Day 7 of Girlscript Education Outreach program, Batch 5.
3.3 Logarithmic Functions
This slide show give the details of how to find the domain of a logarithmic function, graph it using the MyLabsPlus graphing software, and then find the x and y intercepts of that logarithmic function.
Pre-Cal 40S Slides April 7, 2008
This document provides an introduction to exponential functions by explaining that exponential functions involve raising a base number to a power. It defines the key terms in an exponential expression, including the base, which is the number being raised; the exponent, which tells how many times the base is used as a factor; and the power, which is the result of the exponential expression.
Lecture 27 simulated annealing
Simulated annealing is an algorithm that mimics the physical process of annealing to find the global minimum of a cost function. It works by randomly perturbing the current solution and accepting changes that increase the cost with a probability that decreases over time, allowing it to escape local minima. The algorithm includes a temperature parameter that controls the probability of accepting worse solutions, and lowering it slowly over iterations helps find better solutions than simple hill climbing.
K map
This document describes a K-Map software tool that simplifies Boolean equations. The tool reads in a Boolean expression with up to 4 variables in sum-of-products or product-of-sums form, generates a Karnaugh map, and uses it to minimize the expression. Algorithms are provided for solving 2, 3, and 4 variable maps. The tool could aid in designing sequential circuits and simplifying expressions frequently in other applications. Its use of different input forms and deductive reasoning achieves simplified output.
Recommended
Pre-Cal 40S Slides April 11, 2008
This document discusses logarithms and their relationship to exponents. It states that a logarithm is an exponent, with the base of the logarithm corresponding to the base of the exponential expression. It provides examples of converting between logarithmic and exponential forms using the definitions that the logarithm is the exponent and its argument is the power. The document also presents the product, quotient, and power laws for logarithms as methods for simplifying logarithmic expressions.
Automaton
Definition of an Automaton, characteristics of automata,types of automata,deterministic finite automata,non-deterministic automata,Example of Automata,Graphical representation of DFA and NDFA. transition table and transition diagram of DFA and NDFA
Lecture 11 sections 4.3-4.4 logarithmic functions
This document discusses logarithmic functions and their properties. It begins with examples of converting between exponential and logarithmic form. It then covers the natural logarithm function and using transformations of logarithmic functions. The main section discusses properties of logarithms, including the product rule, quotient rule, and power rule for logarithms. It provides examples of writing logarithmic expressions using these rules in both expanded and condensed form.
Theory of Computation FSM Grammar Minimisation and Normal Forms
This was the content discussed for Theory of Computation on Day 7 of Girlscript Education Outreach program, Batch 5.
3.3 Logarithmic Functions
This slide show give the details of how to find the domain of a logarithmic function, graph it using the MyLabsPlus graphing software, and then find the x and y intercepts of that logarithmic function.
Pre-Cal 40S Slides April 7, 2008
This document provides an introduction to exponential functions by explaining that exponential functions involve raising a base number to a power. It defines the key terms in an exponential expression, including the base, which is the number being raised; the exponent, which tells how many times the base is used as a factor; and the power, which is the result of the exponential expression.
Lecture 27 simulated annealing
Simulated annealing is an algorithm that mimics the physical process of annealing to find the global minimum of a cost function. It works by randomly perturbing the current solution and accepting changes that increase the cost with a probability that decreases over time, allowing it to escape local minima. The algorithm includes a temperature parameter that controls the probability of accepting worse solutions, and lowering it slowly over iterations helps find better solutions than simple hill climbing.
K map
This document describes a K-Map software tool that simplifies Boolean equations. The tool reads in a Boolean expression with up to 4 variables in sum-of-products or product-of-sums form, generates a Karnaugh map, and uses it to minimize the expression. Algorithms are provided for solving 2, 3, and 4 variable maps. The tool could aid in designing sequential circuits and simplifying expressions frequently in other applications. Its use of different input forms and deductive reasoning achieves simplified output.
AP Calculus Slides December 12, 2007
The document discusses properties of integrals including:
1) The Fundamental Theorem of Calculus which states that integrating the derivative of a function over an interval gives the total change in the function over that interval.
2) Rules for integrals including the additive interval rule, constant multiple rule, sum and difference rule, and inequality rule.
3) Examples are provided to illustrate applying these rules to evaluate integrals.
Polish Notation In Data Structure
The document discusses different notation styles for representing arithmetic expressions, including infix, prefix, and postfix notations. It provides examples of converting expressions between these notations. Infix notation is the conventional style that humans use, but prefix and postfix notations are better suited for computer parsing. The document also covers parsing expressions, operator precedence, and the steps to convert between infix and prefix and infix and postfix notations.
CNF & Leftmost Derivation - Theory of Computation
Theory of Computation- Chomsky Normal Form & Leftmost Derivation - 1CNF - 2CNF - 3CNF - BCNF and Examples
D igl elex-day5_recorded-lec
The document discusses logic circuits and summarizes key concepts including:
1. Karnaugh maps are used to minimize Boolean expressions without using algebra by grouping adjacent 1's and X's. They are useful for expressions with 2-4 variables.
2. Truth tables and sum-of-products (SOP) form are introduced to represent logic expressions. Examples show converting between truth tables, SOP expressions, and logic circuits.
3. A worked example of a logic circuit is presented to detect conditions for an automobile alarm based on door, ignition, and headlight switches.
4. Objectives of an experiment on combinational logic circuits are outlined, including deriving and simplifying logic equations, converting
Dynamic programming for simd
This document discusses using SIMD (Single Instruction Multiple Data) instructions to parallelize dynamic programming algorithms. It provides an example of how a SIMD register can execute the same operation on multiple data elements in parallel. It also describes how to partition the dynamic programming problem into properly aligned chunks that can be processed simultaneously using SIMD to improve performance. Boundary conditions and dependencies between chunks are identified as challenges to be addressed for an effective parallel implementation.
Tma2033 chap2.1(binary op)handout
This document defines and provides examples of binary operations, properties of operations like commutativity and associativity, and special elements related to operations like identity and inverse elements. A binary operation on a set A is a rule that assigns exactly one element a*b in A to each ordered pair (a,b) of elements in A. Examples of properties discussed are an operation being commutative if a*b = b*a for all a,b in A, and associative if (a*b)*c = a*(b*c) for all a,b,c in A. Identity elements satisfy e*a = a*e = a for all a, and inverse elements satisfy a*x = e = x
Linear algebraweek2
The document discusses row operations and reduced row-echelon form. There are three types of row operations that can be performed on a matrix: swapping rows, multiplying a row by a nonzero number, and multiplying a row by a number and adding it to another row. Reduced row-echelon form has specific properties: rows of all zeros must be below non-zero rows, the leftmost non-zero entry in a row must be 1 and the only non-zero in its column, and leftmost non-zero entries must be in ascending order of row and column.
College Algebra 5.4
This document discusses solving exponential and logarithmic equations. It notes that:
1) Logarithmic equations can be solved using the one-to-one property by setting log expressions equal.
2) Exponential equations can be transformed into logarithmic form by taking the log of both sides, then solved for the variable.
3) There are several steps to solve logarithmic equations - combining log terms, using one-to-one properties, or converting to exponential form.
Scheme language brief introduction
This document discusses the programming language Scheme. It outlines that Scheme is a functional programming language that emphasizes minimalism and uses lambda calculus. It has no loops like "for" or "while", instead using recursion. The document also explains how Scheme can be used to represent mathematical concepts like lambda calculus. Finally, it mentions Scheme's ability to implement itself through "metalinguistic abstraction".
Problem set2 | Theory of Computation | Akash Anand | MTH 401A | IIT Kanpur
This document contains 20 problems related to theory of computation and context-free languages. The problems cover topics such as constructing grammars and pushdown automata for specific languages, proving languages are/aren't context-free, ambiguity of grammars, closure properties of context-free languages, and decidability questions.
Math Stuff
The document discusses properties and definitions related to exponents, logarithms, and exponential and logarithmic functions. It defines exponent properties like the product, quotient, power, and negative exponent properties. It also defines logarithms as the inverse of exponential functions and discusses properties of logarithms like the product, power, and change of base properties. Rational exponents and their relationship to roots is explained. The general forms of exponential and power functions are provided.
A27 3 powerfxns
This document discusses function operations and power functions in algebra. It states that functions can be added, subtracted, multiplied, and divided to create new functions. It also explains that power functions have the form y=ax^b, where b is the exponent, and if b is an integer the function is a polynomial function. Additionally, it notes that composition of functions creates a new function and can be written as f(g(x)).
Formulae
This document discusses logarithmic and exponential functions as well as trigonometric identities, functions, and their derivatives and integrals. It covers topics such as f(x)=sin x, f(x)=cos x, f(x)=tan x, and trigonometric identities as well as functions and their derivatives and integrals.
To know test 11
The document provides an overview of mathematical concepts to study for a test, including abstract cost problems, inverse functions, graphing exponentials, the equation of a line equidistant from two points, sinusoidal functions, permutations, logarithms, trigonometric functions, their inverses, domains, summation notation, the equation of a circle, proofs, radians, the midpoint formula, sketching a function and its reciprocal, and the relationship between radius, arc length, and radians.
Matlab calculus
This document discusses calculus concepts in MATLAB, including:
1) MATLAB can solve problems in differential, integral calculus and differential equations through functions like limit and symbolic math capabilities.
2) It covers calculating limits, derivatives, maxima/minima, and integrals. Examples are provided to calculate one-sided and basic limits.
3) Verification of limit properties is demonstrated using two functions and showing left, right, addition, subtraction, multiplication and division properties hold.
Lesson 12 derivative of inverse trigonometric functions
The document discusses differentiation formulas for inverse trigonometric functions. It defines inverse trigonometric functions and provides the general differentiation formulas for the derivatives of inverse sine, cosine, tangent, cotangent, secant and cosecant functions. Examples are included to demonstrate finding the derivatives of various inverse trigonometric functions using these formulas. Exercises are also provided to allow readers to practice differentiation of inverse trigonometric functions.
Scribe Post 6
The document discusses techniques for integrating complex functions using u-substitution, integration by parts, and trigonometric substitutions. It provides examples of each technique and reminds the reader about LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) for determining the correct approach. The document also emphasizes remembering substitution variables and distributions of coefficients after integration.
Context free grammar
The document provides information about context free grammar (CFG). It defines a CFG as G=(V,T,P,S) where V is the set of nonterminals, T is the set of terminals, P is the set of production rules, and S is the start symbol. Examples of CFGs are provided. Derivation trees, which show the derivation of strings from a CFG, are also discussed. The key differences between regular grammars and CFGs are summarized. Methods for minimizing CFGs by removing useless symbols, epsilon productions, and unit productions are outlined.
Final project Integration
This document discusses integration and properties of integrals. It contains sections on definite integrals, integral properties in two parts, and two tables of indefinite integrals that contain formulas with constants to represent anti-derivatives. The document provides information on the fundamentals of integration through sections and tables.
Pre-Cal 40S Slides April 10, 2008
The document provides instructions for graphing functions by having the reader find the domain and range, horizontal asymptote, x- and y-intercepts, and sketching the graph. It also discusses rewriting tables of values for the inverses of functions and defines logarithms as exponents with equivalent bases and arguments.
Logarithms- principal properties
Logarithms are the inverse of exponential functions. A logarithm represents the power to which a base number must be raised to equal the output number. Logarithmic functions take the form y=logb(x), where b is the base, x is the output number, and y is the exponent or power. Common bases include base 10 and base e. Logarithmic functions have properties that allow logarithms with the same base to be added, subtracted, and multiplied according to certain rules. These properties can be used to solve exponential equations.
More Related Content
What's hot
AP Calculus Slides December 12, 2007
The document discusses properties of integrals including:
1) The Fundamental Theorem of Calculus which states that integrating the derivative of a function over an interval gives the total change in the function over that interval.
2) Rules for integrals including the additive interval rule, constant multiple rule, sum and difference rule, and inequality rule.
3) Examples are provided to illustrate applying these rules to evaluate integrals.
Polish Notation In Data Structure
The document discusses different notation styles for representing arithmetic expressions, including infix, prefix, and postfix notations. It provides examples of converting expressions between these notations. Infix notation is the conventional style that humans use, but prefix and postfix notations are better suited for computer parsing. The document also covers parsing expressions, operator precedence, and the steps to convert between infix and prefix and infix and postfix notations.
CNF & Leftmost Derivation - Theory of Computation
Theory of Computation- Chomsky Normal Form & Leftmost Derivation - 1CNF - 2CNF - 3CNF - BCNF and Examples
D igl elex-day5_recorded-lec
The document discusses logic circuits and summarizes key concepts including:
1. Karnaugh maps are used to minimize Boolean expressions without using algebra by grouping adjacent 1's and X's. They are useful for expressions with 2-4 variables.
2. Truth tables and sum-of-products (SOP) form are introduced to represent logic expressions. Examples show converting between truth tables, SOP expressions, and logic circuits.
3. A worked example of a logic circuit is presented to detect conditions for an automobile alarm based on door, ignition, and headlight switches.
4. Objectives of an experiment on combinational logic circuits are outlined, including deriving and simplifying logic equations, converting
Dynamic programming for simd
This document discusses using SIMD (Single Instruction Multiple Data) instructions to parallelize dynamic programming algorithms. It provides an example of how a SIMD register can execute the same operation on multiple data elements in parallel. It also describes how to partition the dynamic programming problem into properly aligned chunks that can be processed simultaneously using SIMD to improve performance. Boundary conditions and dependencies between chunks are identified as challenges to be addressed for an effective parallel implementation.
Tma2033 chap2.1(binary op)handout
This document defines and provides examples of binary operations, properties of operations like commutativity and associativity, and special elements related to operations like identity and inverse elements. A binary operation on a set A is a rule that assigns exactly one element a*b in A to each ordered pair (a,b) of elements in A. Examples of properties discussed are an operation being commutative if a*b = b*a for all a,b in A, and associative if (a*b)*c = a*(b*c) for all a,b,c in A. Identity elements satisfy e*a = a*e = a for all a, and inverse elements satisfy a*x = e = x
Linear algebraweek2
The document discusses row operations and reduced row-echelon form. There are three types of row operations that can be performed on a matrix: swapping rows, multiplying a row by a nonzero number, and multiplying a row by a number and adding it to another row. Reduced row-echelon form has specific properties: rows of all zeros must be below non-zero rows, the leftmost non-zero entry in a row must be 1 and the only non-zero in its column, and leftmost non-zero entries must be in ascending order of row and column.
College Algebra 5.4
This document discusses solving exponential and logarithmic equations. It notes that:
1) Logarithmic equations can be solved using the one-to-one property by setting log expressions equal.
2) Exponential equations can be transformed into logarithmic form by taking the log of both sides, then solved for the variable.
3) There are several steps to solve logarithmic equations - combining log terms, using one-to-one properties, or converting to exponential form.
Scheme language brief introduction
This document discusses the programming language Scheme. It outlines that Scheme is a functional programming language that emphasizes minimalism and uses lambda calculus. It has no loops like "for" or "while", instead using recursion. The document also explains how Scheme can be used to represent mathematical concepts like lambda calculus. Finally, it mentions Scheme's ability to implement itself through "metalinguistic abstraction".
Problem set2 | Theory of Computation | Akash Anand | MTH 401A | IIT Kanpur
This document contains 20 problems related to theory of computation and context-free languages. The problems cover topics such as constructing grammars and pushdown automata for specific languages, proving languages are/aren't context-free, ambiguity of grammars, closure properties of context-free languages, and decidability questions.
Math Stuff
The document discusses properties and definitions related to exponents, logarithms, and exponential and logarithmic functions. It defines exponent properties like the product, quotient, power, and negative exponent properties. It also defines logarithms as the inverse of exponential functions and discusses properties of logarithms like the product, power, and change of base properties. Rational exponents and their relationship to roots is explained. The general forms of exponential and power functions are provided.
A27 3 powerfxns
This document discusses function operations and power functions in algebra. It states that functions can be added, subtracted, multiplied, and divided to create new functions. It also explains that power functions have the form y=ax^b, where b is the exponent, and if b is an integer the function is a polynomial function. Additionally, it notes that composition of functions creates a new function and can be written as f(g(x)).
Formulae
This document discusses logarithmic and exponential functions as well as trigonometric identities, functions, and their derivatives and integrals. It covers topics such as f(x)=sin x, f(x)=cos x, f(x)=tan x, and trigonometric identities as well as functions and their derivatives and integrals.
To know test 11
The document provides an overview of mathematical concepts to study for a test, including abstract cost problems, inverse functions, graphing exponentials, the equation of a line equidistant from two points, sinusoidal functions, permutations, logarithms, trigonometric functions, their inverses, domains, summation notation, the equation of a circle, proofs, radians, the midpoint formula, sketching a function and its reciprocal, and the relationship between radius, arc length, and radians.
Matlab calculus
This document discusses calculus concepts in MATLAB, including:
1) MATLAB can solve problems in differential, integral calculus and differential equations through functions like limit and symbolic math capabilities.
2) It covers calculating limits, derivatives, maxima/minima, and integrals. Examples are provided to calculate one-sided and basic limits.
3) Verification of limit properties is demonstrated using two functions and showing left, right, addition, subtraction, multiplication and division properties hold.
Lesson 12 derivative of inverse trigonometric functions
The document discusses differentiation formulas for inverse trigonometric functions. It defines inverse trigonometric functions and provides the general differentiation formulas for the derivatives of inverse sine, cosine, tangent, cotangent, secant and cosecant functions. Examples are included to demonstrate finding the derivatives of various inverse trigonometric functions using these formulas. Exercises are also provided to allow readers to practice differentiation of inverse trigonometric functions.
Scribe Post 6
The document discusses techniques for integrating complex functions using u-substitution, integration by parts, and trigonometric substitutions. It provides examples of each technique and reminds the reader about LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) for determining the correct approach. The document also emphasizes remembering substitution variables and distributions of coefficients after integration.
Context free grammar
The document provides information about context free grammar (CFG). It defines a CFG as G=(V,T,P,S) where V is the set of nonterminals, T is the set of terminals, P is the set of production rules, and S is the start symbol. Examples of CFGs are provided. Derivation trees, which show the derivation of strings from a CFG, are also discussed. The key differences between regular grammars and CFGs are summarized. Methods for minimizing CFGs by removing useless symbols, epsilon productions, and unit productions are outlined.
Final project Integration
This document discusses integration and properties of integrals. It contains sections on definite integrals, integral properties in two parts, and two tables of indefinite integrals that contain formulas with constants to represent anti-derivatives. The document provides information on the fundamentals of integration through sections and tables.
What's hot (19)
Problem set2 | Theory of Computation | Akash Anand | MTH 401A | IIT Kanpur
Problem set2 | Theory of Computation | Akash Anand | MTH 401A | IIT Kanpur
Lesson 12 derivative of inverse trigonometric functions
Lesson 12 derivative of inverse trigonometric functions
Similar to Pre-Cal 40S April 17, 2009
Pre-Cal 40S Slides April 10, 2008
The document provides instructions for graphing functions by having the reader find the domain and range, horizontal asymptote, x- and y-intercepts, and sketching the graph. It also discusses rewriting tables of values for the inverses of functions and defines logarithms as exponents with equivalent bases and arguments.
Logarithms- principal properties
Logarithms are the inverse of exponential functions. A logarithm represents the power to which a base number must be raised to equal the output number. Logarithmic functions take the form y=logb(x), where b is the base, x is the output number, and y is the exponent or power. Common bases include base 10 and base e. Logarithmic functions have properties that allow logarithms with the same base to be added, subtracted, and multiplied according to certain rules. These properties can be used to solve exponential equations.
Module 4 exponential and logarithmic functions
This document provides an overview of logarithmic functions including:
1) Logarithmic functions are defined as the inverse of exponential functions.
2) The module will cover defining logarithmic functions, graphing them, applying logarithmic laws, and solving logarithmic equations.
3) Students are expected to learn to define logarithmic functions, draw their graphs, describe properties from graphs, apply laws of logarithms, and solve simple logarithmic equations.
Math130 ch09
This document provides an overview of exponential and logarithmic functions. It covers composite and inverse functions, exponential functions, logarithmic functions, properties of logarithms, common logarithms, exponential and logarithmic equations, and natural exponential and logarithmic functions. Example problems are provided to illustrate each concept.
Properties of-logarithms
1. Logarithmic functions are the inverse of exponential functions. The logarithm of a number represents the power to which the base must be raised to produce that number.
2. Common properties of logarithms include adding exponents when multiplying terms with the same base and subtracting exponents when dividing terms with the same base.
3. The change of base formula allows converting between logarithms with different bases using a ratio of logarithms and the natural logarithm.
Pre-Cal 40S April 14, 2009
John Napier was a 16th century Scottish mathematician and philosopher known for his discovery of logarithms. Logarithms allow for easier calculations by converting multiplications and divisions into additions and subtractions. The document discusses exponential functions and their inverses, and defines logarithms as the inverse of exponential functions, where the exponent becomes the logarithm, the base stays the same, and the power becomes the argument.
Module 4 exponential and logarithmic functions
This document provides an overview of a module on logarithmic functions. It discusses the definition of logarithmic functions as the inverse of exponential functions, how to graph logarithmic functions by reflecting the graph of the corresponding exponential function across the line y=x, and properties of logarithmic function graphs like their domains, ranges, asymptotes, and behavior. It also covers laws of logarithms and how to solve logarithmic equations. The document is designed to teach students to define logarithmic functions, graph them, use laws of logarithms, and solve simple logarithmic equations.
8.4 logarithmic functions
Logarithms were invented to solve exponential equations like 2x = 6, where x is between 2 and 3. A logarithm with base b is defined as logby = x if and only if bx = y. Common properties of logarithms include logb1 = 0 and logbb = 1. Logarithmic functions have inverse relationships with exponential functions and can be graphed by shifting and reflecting the standard logarithmic curve.
Logarithmic Functions
The document defines logarithmic functions and provides examples of converting between logarithmic and exponential forms. It discusses the properties of logarithmic equality and solving logarithmic equations by rewriting them in exponential form. Key points include: logarithmic and exponential forms are inverses; the exponent becomes the logarithm; logarithmic functions have a domain of positive real numbers; and solving logarithms involves setting arguments equal when bases are the same.
Exponents and Logs
This document covers exponents, logarithms, and their functions. It defines exponential and logarithmic expressions, reviews rules for manipulating exponents and logarithms, and examines the graphs of exponential and logarithmic functions. It provides examples of applying exponent and logarithm rules, and contains activities involving evaluating exponential and logarithmic expressions and graphing their related functions.
1. Graph the exponential function by hand. Identify any asymptotes.docx
1. Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. (Enter NONE in any unused answer blanks.)
Equation of horizontal asymptote:
Equation of vertical asymptote:
Value of y-intercept
Value of x-intercept
The function is
2. Use the graph of y = 2x to match the function with its graph.
A
B
C
D
y = 2x – 4
y = 2x – 5
y = 2x + 4
y = 2–x
3. Use the graph of f to describe the transformation that yields the graph of g. Then sketch the graphs of f and g by hand.
f(x) = −2x, g(x) = 5 − 2x
The graph of g(x) = 5 − 2x is a vertical shift five units downward of f(x) = −2x.
The graph of g(x) = 5 − 2x is a horizontal shift five units to the left of f(x) = −2x.
The graph of g(x) = 5 − 2x is a vertical shift five units upward of f(x) = −2x.
The graph of g(x) = 5 − 2x is a horizontal shift five units to the right of f(x) = −2x.
Sketch the graphs of f and g.
4. Use a graphing utility to construct a table of values for the function. (Round your answers to three decimal places.)
x
f(x) = 5x − 3
-1
0
1
2
3
Sketch the graph of the function.
Identify any asymptotes of the graph. (Enter NONE in any unused answer blanks.)
vertical asymptote
x
=
horizontal asymptote
y
=
5. Use a graphing utility to construct a table of values for the function. (Round your answers to three decimal places.)
x
g(x) = 4 − e−3x
-4
-3
-2
-1
0
Sketch the graph of the function.
Identify any asymptotes of the graph. (Enter NONE in any unused answer blanks.)
vertical asymptote
x
=
horizontal asymptote
y
=
6. Fill in the blank.
If
x = ey,
then y = .
7. For what value of x is
ln x = ln 9?
x =
8. Write the logarithmic equation in exponential form. For example, the exponential form of
log5 25 = 2 is 52 = 25.
log2 512 = 9
=
9. Write the logarithmic equation in exponential form. For example, the exponential form of log5 25 = 2 is 52 = 25.
log
1
100,000,000
= -8
=
10. Write the exponential equation in logarithmic form. For example, the logarithmic form of
23 = 8 is log2(8) = 3.
43/2 = 8
log
=
11. Use a calculator to evaluate the function at the indicated value of x. Round your result to three decimal places.
f(x) = log10(x) x = 4/5
12. Solve the equation for x.
log10(102) = x
13. Write the logarithmic equation in exponential form. For example, the exponential form of ln(5) = 1.6094... is e1.6094... = 5. (Do not use ... in your answer.)
=
14. Write the exponential equation in logarithmic form. For example, the logarithmic form of
e2 = 7.3890 is ln 7.3890 = 2.
(Do not use ... in your answer.)
e2.2 = 9.0250
ln
=
15. Use the properties of natural logarithms to rewrite the expression.
5 ln(e5)
16. Use the properties of logarithms to rewrite and simplify the logarithmic expression.
ln
9
e9
17. Use the properties of logarithms to expand the expression as a ...
2006 : Solving Linear Problems with MOSEK (Pittsburgh 2006)
The document discusses recent improvements to the simplex algorithm implemented in MOSEK. It describes how MOSEK exploits sparsity aggressively when solving systems involving the basis factorization. It also discusses work to improve numerical stability, such as updating the LU factorization before updating solutions. Additionally, it covers enhancements to pricing strategies for choosing entering columns, including partial pricing, steepest edge pricing, and approximations.
Business Math Chapter 2
This document introduces exponents, logarithms, and exponential and logarithmic equations. It defines exponents and logarithms, including common, natural, and base conversion. Laws of exponents and logarithms are presented along with properties of radicals, rational exponents, negative exponents, and zero exponents. Examples of solving exponential and logarithmic equations are provided. The summary provides an overview of the key topics and concepts covered in the document.
090601 logs
Logarithms are the inverse function of exponents. A logarithm asks "what power of the base yields the input value?". Common logarithms have a base of 10. The change-of-base formula allows converting between logarithms of different bases. Logarithm properties include log(uv)=log(u)+log(v) and log(u/v)=log(u)-log(v). Condensing and expanding logarithms uses these properties to combine or separate logarithms. Setting logarithms of the same base equal allows solving exponential equations.
Bab 4 dan bab 5 algebra and trigonometry
This document discusses exponential functions and their properties. Exponential functions have the form f(x) = ax, where a is the base. They model phenomena like population growth, compound interest, and radioactive decay. Exponential functions either increase or decrease rapidly depending on whether the base a is greater than or less than 1. Their graphs have a characteristic shape, with the x-axis as a horizontal asymptote. Logarithmic functions are the inverses of exponential functions and can be used to solve equations involving exponential models.
4.3 Logarithmic Functions
This document provides an overview of logarithmic functions. It defines logarithms as the inverse of exponents, where the logarithm of a number is the exponent that the base must be raised to to produce that number. It presents examples of solving logarithmic equations by rewriting them as exponential equations. The key properties of logarithms - that they satisfy the same properties as exponents - are described. The document works through multiple examples of evaluating logarithms and using their properties to simplify expressions and solve equations.
Functional Concepts for OOP Developers
The document provides an overview of functional programming concepts including:
- Functional languages like Haskell, Scala, Clojure, F#, Erlang, and Lisp/Scheme
- Concepts of immutability, concurrency, side effects, and using monads to deal with side effects
- Examples demonstrating functional programming techniques like pattern matching, immutable collections, message passing actors, and software transactional memory (STM)
Logarithmic functions (2)
Logarithmic functions relate the exponent of a number to the logarithm of its base. They are inverse functions of exponentials, where the domain and range are flipped. Key properties include:
1) Logarithmic graphs have a domain of positive real numbers and range of all real numbers, while being always increasing with a vertical asymptote at y=0.
2) Logarithms can have various bases, with common bases being base 10 (common log), base e (natural log), and base 2 or 10.
3) Logarithmic equations can be solved by converting them to exponential form and then solving the exponential equation.
Teoria y problemas de logaritmos l5 ccesa007
1. The document defines logarithms as the exponent to which a base must be raised to obtain a given number. It also discusses the fundamental properties and identities of logarithms.
2. The fundamental properties include: the logarithm of 1 is 0; the logarithm of the base is 1; the logarithm of a product is the sum of the logarithms of the factors; the logarithm of a quotient is the logarithm of the dividend minus the logarithm of the divisor.
3. The document also discusses cologarithms, antilogarithms, and provides examples of solving logarithmic equations.
Similar to Pre-Cal 40S April 17, 2009 (20)
1. Graph the exponential function by hand. Identify any asymptotes.docx
1. Graph the exponential function by hand. Identify any asymptotes.docx
2006 : Solving Linear Problems with MOSEK (Pittsburgh 2006)
2006 : Solving Linear Problems with MOSEK (Pittsburgh 2006)
More from Darren Kuropatwa
Behind Their Eyes v1
Behind Their Eyes - making thinking visible is not enough
Walk into any classroom and watch the breakneck pace at which teachers are working hard to help students learn. Mind you, if we don’t uncover what students are thinking while learning, they may be running down the wrong path. OK, so we need ways to make student thinking visible. Seeing their thinking is important, but we also need to create the time and space for teachers to absorb, reflect, and act on what their students thinking reveals. This workshop shares strategies both for making student thinking visible and for creating time and space for teachers to meaningfully act on what they learn about what’s going on behind their eyes.
Leading Change v1
“If you really want to understand something, try changing it.” - Kurt Lewin
As the Director of Learning for a school division made up of 18 schools, my job is to help lead the largest change initiative ever undertaken in our school community. One of the most important, difficult, messy things any school leader does is lead change. While we can learn from the change leadership of others, copying their work most often leads to failure. Success is more likely to come from adapting others work to our own context. In this workshop I share the journey we’ve undertaken collectively in our schools; how we developed a shared vision, cultivated collaborative cultures, maintained a focus on deep learning, and wrestle with the nuances of accountability. Informed by the latest research on change management in education, we also model strategies for fostering deep learning conversations in your schools. We’ll engage in some deeper learning conversations together and take back a wealth of ideas you can adapt to your own context. Developing collaborative cultures is careful and precise work that has profound impact when carried out well. So how do you do that? Come, let’s learn together. Good people are important, but good cultures are moreso.
Providing Permission To Wonder v2.1
In a world where knowledge is more a verb than a noun how do we foster deep learning in our students? Good questions cause thinking. Unfortunately, many of the questions regularly asked in classrooms focus on knowledge as a noun. This presentation will explore inquiry as a pedagogical stance and the effective use of thinking and learning tools in the classroom. We will work together to model teaching practices that lead to students co-constructing a networked (real world) rather than hierarchical (artificial) understanding of their world regardless of grade level or discipline.
Participants will leave this workshop with a toolkit of research based questioning and thinking strategies they can begin using with their students tomorrow.
Things That Suck About Digital Citizenship v1
The document is a presentation about digital citizenship given by Darren Kuropatwa at the Building Learning Communities Conference in Boston, MA in July 2017. It discusses the importance of digital citizenship and responding to adversity with persistent kindness. It provides examples of digital citizenship issues and scenarios for discussion. It encourages participants to think about their own digital footprint and how to be good digital citizens.
Digital Storytelling for Deeper Learning v1
Presented at the Riding the Wave Conference in Gimli, Manitoba. May 2017.
In two words, you remember the whole story: glass slipper, sour grapes, cold porridge. You remember more than facts, you recall relationships & deeper connections between characters. Some of the powerful ways we leverage digital for deeper learning includes challenging sources of information (fake news), exploring bias (developing empathy through multiple perspectives), and creating powerful feedback loops that foster deeper learning.
Powerful narratives, in a word or two, bring to mind a wealth of ideas & relationships; more than just facts. How can we find stories that make our teaching sticky and help kids find, and more importantly tell, stories that make learning stick? This workshop will equip teachers with the skills & knowledge to foster deeper learning across the curriculum by intentionally leveraging digital tools to foster deeper learning.
Tales of Learning and the Gifts of Footprints v4.2
This document appears to be a presentation about digital learning and storytelling. It discusses shifting from compliance to care, private to public learning, and consumer to participatory models. It addresses what digital storytellers look like and principles of learning including starting where students are, learning being done by and for students, students talking about learning, having learning targets, and feedback. It encourages generosity, sharing tales of learning, and giving the gifts of footprints.
The Fourth Screen v4.2
Presented at the Richmond District Conference, Feb 2017.
A series of stories woven together to start a conversation with middle and high school students, teachers, and parents about living our lives on and offline (on The Fourth Screen) more thoughtfully.
This talk focuses primarily on the ideas of Empathy, Empowerment & Persistent Kindness and shares resources teachers can use to lead these sorts of conversations with their own students.
Making Student Thinking Visible v4
Slides to support a master class on making student thinking visible through practical hands-on activities and structured around Dylan Wiliam's work on formative assessment and active learning. Held at the BYTE Conference 2017 in Portage la Prairie, Manitoba.
Learning is at BYTE 2017
A group of educators from the BYTE Conference 2017 (Build Your Teaching Experience) share their ideas about learning as a series of visual metaphors they found on their phones.
Leveraging Digital for Deeper Learning
Slides to support a Keynote presentation to Early Years and Middle Years teachers in Hanover School Division on 3 February 2017.
We Learn Through Stories at PRIZMAH17
The document discusses storytelling and how it can be used as a tool for learning. It suggests that storytelling allows students to think in metaphors and learn through stories. It provides examples of how digital tools like QR codes and apps can be used to incorporate storytelling into the classroom. It also outlines some rules of thumb for using storytelling, such as personalizing tasks to students' experiences, collaborating on group projects, and getting feedback from both inside and outside the classroom.
Providing Permission to Wonder v3
In a world where knowledge is more a verb than a noun how do we foster deep learning in our students? Good questions cause thinking. Unfortunately, many of the questions regularly asked in classrooms focus on knowledge as a noun. This presentation will explore the effective use of thinking and learning tools in the classroom. We will work together to model teaching practices that lead to students co-constructing a networked (real world) rather than hierarchical (artificial) understanding of their world regardless of grade level or discipline.
Participants leave this workshop with a toolkit of research based questioning and thinking strategies they can begin using with their students tomorrow.
The Fourth Screen v4.1
This document contains multiple sections on topics related to technology and its impact on society, including how the internet allows information to be easily shared but also persist indefinitely, issues around online privacy and bullying, and ways for parents to support their children's safe and responsible internet use. The document advocates for empowering youth and promoting kindness both online and off.
Making Student Thinking Visible v3.7
Slides to support a master class on making student thinking visible through practical hands-on activities and structured around Dylan Wiliam's work on formative assessment and active learning.
Learning is at AUHSD
A group of educators from the Anderson Union High School & Redding School Districts and share their ideas about learning as a series of visual metaphors.
Providing Permission to Wonder v2
In a world where knowledge is more a verb than a noun how do we foster deep learning in our students? Good questions cause thinking. Unfortunately, many of the questions regularly asked in classrooms focus on knowledge as a noun. This presentation will explore the effective use of thinking and learning tools in the classroom. We will work together to model teaching practices that lead to students co-constructing a networked (real world) rather than hierarchical (artificial) understanding of their world regardless of grade level or discipline.
Participants leave this workshop with a toolkit of research based questioning and thinking strategies they can begin using with their students tomorrow.
The Fourth Screen v4
This document contains a collection of images, quotes, and short passages on topics related to online communities, sharing, and empowerment through technology. The snippets discuss how the internet allows information to be easily shared, encourages learning, and can help empower victims of bullying. The overarching theme is about the positive impact community and connection through online platforms can provide.
Learning is at BLC16
A group of educators at the Building Learning Communities Conference share their ideas about learning as a series of visual metaphors.
We Learn Through Stories v4 (master class)
Slides to support a master class at the Building Learning Communities Conference in Boston, MA. 18 July 2016.
How can we make learning sticky using powerful storytelling frameworks that tap into peoples' emotions? How do we involve all students in creating digital content that doesn't also create hours of content for teachers to assess? This interactive session will showcase Digital Storytelling activities teachers can use in class tomorrow! Document student learning & foster reflective ways for students to share their learning. 1st: we play! Then we'll discuss how to practically adapt these ideas, make them your own, and figure out what sort of infrastructure needs to be in place to support these kinds of powerful learning experiences. We’ll learn how to exercise your students' & your own creativity muscles and share simple strategies for collecting & publishing student work.
Deep Learning Design: the middle ring
Slides in support of a professional learning day for administrators in Hanover School Division focused on developing a common language & understanding of Deep Learning Design.
More from Darren Kuropatwa (20)
Tales of Learning and the Gifts of Footprints v4.2
Tales of Learning and the Gifts of Footprints v4.2
Recently uploaded
How to Setup Warehouse & Location in Odoo 17 Inventory
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
Hindi varnamala | hindi alphabet PPT.pdf
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
The Diamonds of 2023-2024 in the IGRA collection
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
Chapter 4 - Islamic Financial Institutions in Malaysia.pptxMohd Adib Abd Muin, Senior Lecturer at Universiti Utara Malaysia
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...National Information Standards Organization (NISO)
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.How to Build a Module in Odoo 17 Using the Scaffold Method
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
How to Manage Your Lost Opportunities in Odoo 17 CRM
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
How to Fix the Import Error in the Odoo 17
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Pengantar Penggunaan Flutter - Dart programming language1.pptx
Pengantar Penggunaan Flutter - Dart programming language1.pptx
clinical examination of hip joint (1).pdf
described clinical examination all orthopeadic conditions .
How to Add Chatter in the odoo 17 ERP Module
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
DRUGS AND ITS classification slide share
Any substance (other than food) that is used to prevent, diagnose, treat, or relieve symptoms of a
disease or abnormal condition
Recently uploaded (20)
How to Setup Warehouse & Location in Odoo 17 Inventory
How to Setup Warehouse & Location in Odoo 17 Inventory
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
How to Build a Module in Odoo 17 Using the Scaffold Method
How to Build a Module in Odoo 17 Using the Scaffold Method
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
How to Manage Your Lost Opportunities in Odoo 17 CRM
How to Manage Your Lost Opportunities in Odoo 17 CRM
Pengantar Penggunaan Flutter - Dart programming language1.pptx
Pengantar Penggunaan Flutter - Dart programming language1.pptx
Film vocab for eal 3 students: Australia the movie
Film vocab for eal 3 students: Australia the movie
Pre-Cal 40S April 17, 2009
- 1. Don't forget ... a logarithm is an exponent! ... if you do you may get mud on your face. Pararescue by flickr user slagheap
- 2. Let's see 2^x Then, sketch the graph of it's inverse ...
- 3. Sketch the given function AND it's inverse on the graph.
- 4. Rewrite each table of values for the inverse of each function ...
- 5. The Anatomy Of A Power exponent 3 2 = 8power base power EXPONENTIAL FUNCTIONS turn EXPONENTS into POWERS
- 6. a b=c log b c = a b is the base b is the base a is the exponent a is the (exponent) logarithm c is the power c is the (power) argument These functions are inverses of each other. EXPONENTIAL FUNCTIONS LOGARITHMIC FUNCTIONS turn EXPONENTS into POWERS turn POWERS into EXPONENTS
- 7. a b=c log b c = a b is the base b is the base a is the exponent a is the (exponent) logarithm c is the power c is the (power) argument A Logarithm is an Exponent you're going to forget that
- 8. Graph the function (a) Is there an asymptote? If so state its equation. If not, explain why not. (b) State the domain. (c) State the range. (d) Is the function increasing or decreasing. Explain.
- 9. Simplify . The Product Law
- 11. Simplify The Power Law
- 12. 3 log2 8 = log2 2 = In General ... x log5 5 = x loga a = x x log7 49 = loga a =
- 13. log 39 =9 3 In General ... log ax =x log 710 a = 7