A28 5 log properties notes
•Download as DOCX, PDF•
1 like•120 views
The document discusses properties of logarithms including: 1) The relationship between logbu, logbv, and logbuv follows the product property where logbuv = logbu + logbv. 2) It provides examples of expanding and condensing logarithmic expressions using properties like the product, quotient, and power properties. 3) Calculators typically evaluate common (base 10) and natural (base e) logarithms, and the change of base formula allows converting between different bases.
Report
Share
Report
Share
Recommended
A27 3 powerfxns notes
This document discusses power functions and function operations in algebra 2. It defines power functions as having the form b^x where b is the base and integer exponents result in polynomial functions. It provides examples of adding, subtracting, multiplying, and composing various power functions, finding their domains. Students are assigned homework problems applying these concepts.
1 Rational Expressions Mar 1
This document discusses rational expressions and equations. It defines a rational expression as an algebraic expression that can be written as the quotient of two polynomials. An expression is undefined when its denominator is equal to zero, which are called the non-permissible values. The document provides examples of finding non-permissible values and simplifying rational expressions by factoring and cancelling common factors in the numerator and denominator. It concludes with an assignment to complete practice problems related to rational expressions and equations.
Add math 1
The document contains a mathematics exam with 15 multiple-choice questions related to functions. It includes diagrams of various functions and asks students to identify properties of the functions such as their domain and range, determine function values, and perform operations on composite functions.
7th pre alg -l2
This document contains a math lesson on vocabulary and properties of addition and multiplication. It includes matching vocabulary terms to definitions, listing properties such as commutative and associative, and examples applying the properties. The examples are named by the property they illustrate, such as the identity property of multiplication or commutative property of addition. The document also includes problems to solve and check answers.
8th pre alg -l2
This document contains a math lesson on vocabulary and properties of addition and multiplication. It includes matching vocabulary terms to definitions, listing properties such as commutative and associative, and examples applying the properties. The examples are named by the property they illustrate, such as the identity property of multiplication or commutative property of addition. The document also includes problems to solve and check answers.
A16-5 Graphing Inequalities Notes
To graph a linear inequality in two variables:
1) Graph the corresponding linear equation first as a solid line.
2) For < or >, shade the half-plane on one side of the line. For ≤ or ≥, shade along the line and on one side.
3) Check the given points to see which satisfy the inequality - for the example 2x - 3y ≥ -2, only point (c) satisfies the inequality.
A28-4 log fxns
Logarithmic functions are the inverse of exponential functions. They allow mathematicians to write exponential expressions like 2x = 6 in logarithmic form as log2 6 = x. The graph of a logarithmic function y = logb(x - h) + k will increase if b > 1 and decrease if 0 < b < 1. It will have a vertical asymptote at x = h. Logarithmic functions have the opposite domain and range from their inverse exponential functions.
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.
Recommended
A27 3 powerfxns notes
This document discusses power functions and function operations in algebra 2. It defines power functions as having the form b^x where b is the base and integer exponents result in polynomial functions. It provides examples of adding, subtracting, multiplying, and composing various power functions, finding their domains. Students are assigned homework problems applying these concepts.
1 Rational Expressions Mar 1
This document discusses rational expressions and equations. It defines a rational expression as an algebraic expression that can be written as the quotient of two polynomials. An expression is undefined when its denominator is equal to zero, which are called the non-permissible values. The document provides examples of finding non-permissible values and simplifying rational expressions by factoring and cancelling common factors in the numerator and denominator. It concludes with an assignment to complete practice problems related to rational expressions and equations.
Add math 1
The document contains a mathematics exam with 15 multiple-choice questions related to functions. It includes diagrams of various functions and asks students to identify properties of the functions such as their domain and range, determine function values, and perform operations on composite functions.
7th pre alg -l2
This document contains a math lesson on vocabulary and properties of addition and multiplication. It includes matching vocabulary terms to definitions, listing properties such as commutative and associative, and examples applying the properties. The examples are named by the property they illustrate, such as the identity property of multiplication or commutative property of addition. The document also includes problems to solve and check answers.
8th pre alg -l2
This document contains a math lesson on vocabulary and properties of addition and multiplication. It includes matching vocabulary terms to definitions, listing properties such as commutative and associative, and examples applying the properties. The examples are named by the property they illustrate, such as the identity property of multiplication or commutative property of addition. The document also includes problems to solve and check answers.
A16-5 Graphing Inequalities Notes
To graph a linear inequality in two variables:
1) Graph the corresponding linear equation first as a solid line.
2) For < or >, shade the half-plane on one side of the line. For ≤ or ≥, shade along the line and on one side.
3) Check the given points to see which satisfy the inequality - for the example 2x - 3y ≥ -2, only point (c) satisfies the inequality.
A28-4 log fxns
Logarithmic functions are the inverse of exponential functions. They allow mathematicians to write exponential expressions like 2x = 6 in logarithmic form as log2 6 = x. The graph of a logarithmic function y = logb(x - h) + k will increase if b > 1 and decrease if 0 < b < 1. It will have a vertical asymptote at x = h. Logarithmic functions have the opposite domain and range from their inverse exponential functions.
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.
A1 12 scatter plots
Scatter plots use a series of points to represent a relationship between two variables, with the x-axis representing one variable and the y-axis representing the other. A scatter plot should include a title, labeled x-axis, and labeled y-axis. Correlation can be positive if points increase from left to right, negative if points decrease from left to right, or no correlation if points have no trend. A line of best fit shows the overall trend of the data by having the same number of points above and below the line.
A1 11 functions
The document discusses functions, their domains and ranges, function notation, evaluating functions, and sequences. It provides examples of arithmetic and geometric sequences and how they can be defined using equations. It also discusses determining whether numbers are in the domain of a function and finding the domain and range of functions given their equations or graphs.
A1 11 functions
This document covers functions, including evaluating functions at given inputs, domains and ranges of functions, and sequences. It defines arithmetic and geometric sequences, providing examples of writing equations for each type of sequence and generating additional terms. Students are asked to evaluate functions, write equations for sequences, and find subsequent terms of given arithmetic and geometric sequences.
A2 Test 3 with answers 2011
This study guide covers polynomials, including classifying polynomials by degree and terms, adding and subtracting polynomials, multiplying polynomials, dividing polynomials using long division and synthetic division, factoring quadratics, solving quadratics, and graphing quadratics by finding the vertex, axis of symmetry, and x-intercepts. Key concepts are polynomials having only positive integer exponents, the degree being the highest exponent, and the leading coefficient being the coefficient of the term with the highest exponent.
A1 Test 3 study guide with answers
1. The solution to a system of equations is a point, while the solution to a system of inequalities is a shaded region on a graph.
2. Parallel lines have the same slope and never intersect, so they have no solution. Coinciding lines are the same single line, so they have infinitely many solutions.
3. The document provides examples of solving systems of equations and inequalities through graphing, substitution, and elimination. The solutions are given as points or descriptions of regions.
PC Test 2 study guide 2011
1. The document is a study guide for a precalculus test that covers topics involving vectors, complex numbers, and their representations and operations.
2. It lists 15 problems involving finding magnitudes and direction of vectors, vector components, dot and cross products, complex numbers in rectangular and polar form, and converting between polar and rectangular coordinates.
3. The problems cover basic vector and complex number calculations, representations, properties and conversions.
A2 Test 2 study guide with answers (revised)
The document is an algebra 2 study guide that provides practice problems for graphing and solving various types of inequalities and systems of equations, including:
1) Graphing linear and compound inequalities on number lines and coordinate planes
2) Solving absolute value equations and inequalities
3) Solving systems of linear equations through graphing, substitution, and elimination
The study guide contains over 50 problems to help students prepare for an algebra 2 test.
A2 Test 2 study guide with answers
This document provides a study guide with answers for an Algebra 2 Test 2. It includes instructions to graph and solve various types of inequalities, systems of equations and absolute value equations, including linear and compound inequalities, systems of linear equations solved by graphing, substitution and elimination, and absolute value equations and inequalities.
A1 Test 2 study guide with answers 2011
The document is an algebra 1 test study guide containing 30 problems involving graphing and solving various types of inequalities, including linear, compound, absolute value, and graphing solutions on number lines. Key topics covered include writing inequalities from graphs, solving single-variable and compound inequalities, graphing linear inequalities on a coordinate plane, and solving absolute value equations and inequalities.
A1 Test 2 study guide
1. The document provides a study guide for an Algebra 1 Test 2 with questions about: graphing and writing inequalities on number lines, solving linear inequalities and compound inequalities, graphing linear inequalities on the xy-plane, and solving absolute value equations and inequalities.
2. Questions involve skills like graphing inequalities, writing inequalities from graphs, solving one-step and two-step inequalities, and solving absolute value equations and inequalities.
3. The study guide covers key Algebra 1 concepts to help prepare for Test 2.
PCExam 1 practice with answers
1. This practice exam covers topics like complex numbers, functions, limits, and graphing.
2. It asks students to choose problems involving adding, multiplying, composing, and finding inverses of various functions like f(x)=9x^2+1 and g(x)=x-1.
3. Students also must graph and classify functions, evaluate limits, and perform operations on complex numbers, plotting them on a plane. The exam covers concepts in precalculus.
PCExam 1 study guide answers
1. The document provides examples and definitions for various mathematical concepts including natural numbers, integers, rational numbers, real numbers, imaginary numbers, and complex numbers.
2. It also includes problems involving composition of functions, finding inverses of functions, operations on complex numbers, graphing and classifying functions, and evaluating limits.
3. Examples cover topics like composition and inverses of functions, operations on complex numbers, classifying functions as linear, quadratic, constant, etc. and their domains and ranges, and evaluating limits including ones that are undefined.
PC Exam 1 study guide
This study guide covers topics in precalculus including:
1) Examples of natural numbers, integers, rational numbers, and real numbers.
2) Composition of functions including finding f(g(x)) and g(f(x)) for given functions f(x) and g(x).
3) Finding inverses of functions and verifying inverses by composition.
4) Operations on complex numbers including plotting, finding modulus, distance, midpoint, addition, subtraction, multiplication, and division.
5) Graphing functions, classifying function types, and stating domains and ranges.
6) Finding limits of functions as x approaches values.
PC 1 continuity notes
A continuous function is defined as one that is defined at a point c, where the limit exists and the function approaches the same y-value from both sides of point c. A function may be discontinuous due to an infinite jump, a jump, or a point discontinuity. Rational functions are defined as the quotient of two polynomial functions. They have asymptotes which occur where the denominator is zero, creating either a vertical or horizontal asymptote in the graph. The end behavior of functions can be determined by considering whether they are increasing or decreasing over different intervals.
PC 1 continuity
This document discusses continuity and discontinuities in functions, defining continuous functions as smooth curves where the domain includes all real numbers. It presents tests for continuity, requiring a function to be defined at a point, have an existing function value, and approach the same y-value from both sides. Critical points and extrema are defined as maximums where a function changes from increasing to decreasing, and minimums where it changes from decreasing to increasing. Rational functions are described as the quotient of two polynomial functions, with limited domains and possible vertical asymptotes.
A1 3 linear fxns
The document discusses the slope-intercept form of a linear equation, y=mx+b. It explains that x represents the input, y represents the output, m is the slope of the line, and b is the y-intercept. It provides instructions for graphing a linear equation in slope-intercept form, which is to first plot the y-intercept and then use the slope to rise and run to the next point, connecting the points with a line.
A1 3 linear fxns notes
The document discusses slope and linear equations. It defines slope as rise over run and provides examples of calculating slope from graphs and ordered pairs. It also defines the standard form of a linear equation as y=mx+b, where m is the slope and b is the y-intercept. The document explains how to graph lines from their equations in standard form by plotting the y-intercept and using the slope to rise and run to the next point.
A1 2 linear fxns
This document discusses the slope-intercept form of a linear equation, y=mx+b. It explains that x represents the input, y represents the output, m is the slope of the line, and b is the y-intercept. It provides instructions for graphing a linear equation in slope-intercept form, which is to first plot the y-intercept, then use the slope to rise and run to the next point and connect them with a line.
A1 2 linear fxns notes
This document discusses intercepts and slope of lines in algebra 1. It defines x-intercepts as where the line crosses the x-axis and y-intercepts as where it crosses the y-axis. It provides the equations for finding each, and includes two examples of finding intercepts and graphing lines. It also defines slope as rise over run and asks what the slope is of a given graph, with homework assigned on slope of lines and graphing lines intuitively.
A2 2 linear fxns notes
This document provides instructions and examples for finding the slope and y-intercept of a line from its equation, ordered pairs, or graph and using them to graph the line. It explains that slope is defined as rise over run and is used along with the y-intercept to graph lines by standard form, plotting the y-intercept and using slope to find successive points. Exercises are provided to have students practice finding slope, y-intercept, and graphing lines from different representations.
More Related Content
More from vhiggins1
A1 12 scatter plots
Scatter plots use a series of points to represent a relationship between two variables, with the x-axis representing one variable and the y-axis representing the other. A scatter plot should include a title, labeled x-axis, and labeled y-axis. Correlation can be positive if points increase from left to right, negative if points decrease from left to right, or no correlation if points have no trend. A line of best fit shows the overall trend of the data by having the same number of points above and below the line.
A1 11 functions
The document discusses functions, their domains and ranges, function notation, evaluating functions, and sequences. It provides examples of arithmetic and geometric sequences and how they can be defined using equations. It also discusses determining whether numbers are in the domain of a function and finding the domain and range of functions given their equations or graphs.
A1 11 functions
This document covers functions, including evaluating functions at given inputs, domains and ranges of functions, and sequences. It defines arithmetic and geometric sequences, providing examples of writing equations for each type of sequence and generating additional terms. Students are asked to evaluate functions, write equations for sequences, and find subsequent terms of given arithmetic and geometric sequences.
A2 Test 3 with answers 2011
This study guide covers polynomials, including classifying polynomials by degree and terms, adding and subtracting polynomials, multiplying polynomials, dividing polynomials using long division and synthetic division, factoring quadratics, solving quadratics, and graphing quadratics by finding the vertex, axis of symmetry, and x-intercepts. Key concepts are polynomials having only positive integer exponents, the degree being the highest exponent, and the leading coefficient being the coefficient of the term with the highest exponent.
A1 Test 3 study guide with answers
1. The solution to a system of equations is a point, while the solution to a system of inequalities is a shaded region on a graph.
2. Parallel lines have the same slope and never intersect, so they have no solution. Coinciding lines are the same single line, so they have infinitely many solutions.
3. The document provides examples of solving systems of equations and inequalities through graphing, substitution, and elimination. The solutions are given as points or descriptions of regions.
PC Test 2 study guide 2011
1. The document is a study guide for a precalculus test that covers topics involving vectors, complex numbers, and their representations and operations.
2. It lists 15 problems involving finding magnitudes and direction of vectors, vector components, dot and cross products, complex numbers in rectangular and polar form, and converting between polar and rectangular coordinates.
3. The problems cover basic vector and complex number calculations, representations, properties and conversions.
A2 Test 2 study guide with answers (revised)
The document is an algebra 2 study guide that provides practice problems for graphing and solving various types of inequalities and systems of equations, including:
1) Graphing linear and compound inequalities on number lines and coordinate planes
2) Solving absolute value equations and inequalities
3) Solving systems of linear equations through graphing, substitution, and elimination
The study guide contains over 50 problems to help students prepare for an algebra 2 test.
A2 Test 2 study guide with answers
This document provides a study guide with answers for an Algebra 2 Test 2. It includes instructions to graph and solve various types of inequalities, systems of equations and absolute value equations, including linear and compound inequalities, systems of linear equations solved by graphing, substitution and elimination, and absolute value equations and inequalities.
A1 Test 2 study guide with answers 2011
The document is an algebra 1 test study guide containing 30 problems involving graphing and solving various types of inequalities, including linear, compound, absolute value, and graphing solutions on number lines. Key topics covered include writing inequalities from graphs, solving single-variable and compound inequalities, graphing linear inequalities on a coordinate plane, and solving absolute value equations and inequalities.
A1 Test 2 study guide
1. The document provides a study guide for an Algebra 1 Test 2 with questions about: graphing and writing inequalities on number lines, solving linear inequalities and compound inequalities, graphing linear inequalities on the xy-plane, and solving absolute value equations and inequalities.
2. Questions involve skills like graphing inequalities, writing inequalities from graphs, solving one-step and two-step inequalities, and solving absolute value equations and inequalities.
3. The study guide covers key Algebra 1 concepts to help prepare for Test 2.
PCExam 1 practice with answers
1. This practice exam covers topics like complex numbers, functions, limits, and graphing.
2. It asks students to choose problems involving adding, multiplying, composing, and finding inverses of various functions like f(x)=9x^2+1 and g(x)=x-1.
3. Students also must graph and classify functions, evaluate limits, and perform operations on complex numbers, plotting them on a plane. The exam covers concepts in precalculus.
PCExam 1 study guide answers
1. The document provides examples and definitions for various mathematical concepts including natural numbers, integers, rational numbers, real numbers, imaginary numbers, and complex numbers.
2. It also includes problems involving composition of functions, finding inverses of functions, operations on complex numbers, graphing and classifying functions, and evaluating limits.
3. Examples cover topics like composition and inverses of functions, operations on complex numbers, classifying functions as linear, quadratic, constant, etc. and their domains and ranges, and evaluating limits including ones that are undefined.
PC Exam 1 study guide
This study guide covers topics in precalculus including:
1) Examples of natural numbers, integers, rational numbers, and real numbers.
2) Composition of functions including finding f(g(x)) and g(f(x)) for given functions f(x) and g(x).
3) Finding inverses of functions and verifying inverses by composition.
4) Operations on complex numbers including plotting, finding modulus, distance, midpoint, addition, subtraction, multiplication, and division.
5) Graphing functions, classifying function types, and stating domains and ranges.
6) Finding limits of functions as x approaches values.
PC 1 continuity notes
A continuous function is defined as one that is defined at a point c, where the limit exists and the function approaches the same y-value from both sides of point c. A function may be discontinuous due to an infinite jump, a jump, or a point discontinuity. Rational functions are defined as the quotient of two polynomial functions. They have asymptotes which occur where the denominator is zero, creating either a vertical or horizontal asymptote in the graph. The end behavior of functions can be determined by considering whether they are increasing or decreasing over different intervals.
PC 1 continuity
This document discusses continuity and discontinuities in functions, defining continuous functions as smooth curves where the domain includes all real numbers. It presents tests for continuity, requiring a function to be defined at a point, have an existing function value, and approach the same y-value from both sides. Critical points and extrema are defined as maximums where a function changes from increasing to decreasing, and minimums where it changes from decreasing to increasing. Rational functions are described as the quotient of two polynomial functions, with limited domains and possible vertical asymptotes.
A1 3 linear fxns
The document discusses the slope-intercept form of a linear equation, y=mx+b. It explains that x represents the input, y represents the output, m is the slope of the line, and b is the y-intercept. It provides instructions for graphing a linear equation in slope-intercept form, which is to first plot the y-intercept and then use the slope to rise and run to the next point, connecting the points with a line.
A1 3 linear fxns notes
The document discusses slope and linear equations. It defines slope as rise over run and provides examples of calculating slope from graphs and ordered pairs. It also defines the standard form of a linear equation as y=mx+b, where m is the slope and b is the y-intercept. The document explains how to graph lines from their equations in standard form by plotting the y-intercept and using the slope to rise and run to the next point.
A1 2 linear fxns
This document discusses the slope-intercept form of a linear equation, y=mx+b. It explains that x represents the input, y represents the output, m is the slope of the line, and b is the y-intercept. It provides instructions for graphing a linear equation in slope-intercept form, which is to first plot the y-intercept, then use the slope to rise and run to the next point and connect them with a line.
A1 2 linear fxns notes
This document discusses intercepts and slope of lines in algebra 1. It defines x-intercepts as where the line crosses the x-axis and y-intercepts as where it crosses the y-axis. It provides the equations for finding each, and includes two examples of finding intercepts and graphing lines. It also defines slope as rise over run and asks what the slope is of a given graph, with homework assigned on slope of lines and graphing lines intuitively.
A2 2 linear fxns notes
This document provides instructions and examples for finding the slope and y-intercept of a line from its equation, ordered pairs, or graph and using them to graph the line. It explains that slope is defined as rise over run and is used along with the y-intercept to graph lines by standard form, plotting the y-intercept and using slope to find successive points. Exercises are provided to have students practice finding slope, y-intercept, and graphing lines from different representations.
More from vhiggins1 (20)
A28 5 log properties notes
- 1. Algebra 2<br />8.5Properties of Logarithms<br />logbulogbvlogbuvlog10=log100=log1000=log0.1=log0.01=log0.001=log24=log28=log232=<br />Write a conjecture about the relationship among logbu, logbv, and logbuv.<br />Properties of LogarithmsProduct PropertyQuotient PropertyPower Property<br />Expand log27x3y.Condense log6+2log2-log3.<br />What 2 logarithms does a calculator evaluate? ____________ and ____________<br />Change of Base FormulalogAB= <br />The ____________________________ logarithm has a base b = 10.<br />The ____________________________ logarithm has a base ____________________.<br />Evaluate the expression log3 7 using common and natural logarithms.<br />