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The document discusses properties and laws of logarithms. It provides examples of using logarithm laws to evaluate expressions, such as combining like terms and rewriting expressions as single logarithms. Expressions include adding, subtracting and combining logarithms of various bases applied to expressions involving multiplication, division, powers and variables.
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Day 3 examples u6f13
The document discusses properties and theorems of logarithmic functions including:
1) Logarithmic functions can be combined using addition/subtraction when the bases are the same.
2) Logarithmic functions with composite bases can be rewritten as a single logarithm using laws such as (M·N) = logM + logN.
3) Examples show how to use logarithmic laws to evaluate expressions and combine or simplify logarithmic terms.
Day 3 examples
1) The document discusses logarithmic properties and how to solve logarithmic equations and expressions. It provides examples of evaluating logarithmic expressions using log properties and rewriting expressions as single logarithms.
2) Logarithmic properties and theorems covered include the log property that log(MN) = logM + logN, and rewriting expressions with more than one logarithm as a single logarithm.
3) Examples show how to use these properties to evaluate expressions such as log6 8 + log6 9 - log6 2 and rewrite expressions like (x - 2) - 3logx as a single logarithm.
Exercise #22
This document outlines three theorems for logarithms:
1) The Product Theorem states that the log of a product is equal to the sum of the logs of the factors.
2) The Quotient Theorem states that the log of a quotient is equal to the difference of the logs of the factors.
3) The Power Theorem states that the log of a factor to a given power is equal to the power times the log of the factor.
These theorems only apply when the logarithms have the same base. Examples are provided to demonstrate applying the theorems to simplify logarithmic expressions and evaluate logarithms.
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Log summary & equations
Here are solutions to the practice problems:
log(2x – 4) = 3
2x – 4 = 23 = 8
2x – 4 = 8
2x = 12
x = 6
log2(x) + log2 (x – 6) = 4
log2(x(x – 6)) = 4
x(x – 6) = 24
x2 – 6x – 24 = 0
(x – 6)(x + 4) = 0
x = 6, x = -4
log4(x + 4) – log4(x – 1) = 2
log4((x + 4)/(x – 1)) = 2
(x
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.
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.
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The document discusses properties and theorems of logarithmic functions including:
1) Logarithmic functions can be combined using addition/subtraction when the bases are the same.
2) Logarithmic functions with composite bases can be rewritten as a single logarithm using laws such as (M·N) = logM + logN.
3) Examples show how to use logarithmic laws to evaluate expressions and combine or simplify logarithmic terms.
Day 3 examples
1) The document discusses logarithmic properties and how to solve logarithmic equations and expressions. It provides examples of evaluating logarithmic expressions using log properties and rewriting expressions as single logarithms.
2) Logarithmic properties and theorems covered include the log property that log(MN) = logM + logN, and rewriting expressions with more than one logarithm as a single logarithm.
3) Examples show how to use these properties to evaluate expressions such as log6 8 + log6 9 - log6 2 and rewrite expressions like (x - 2) - 3logx as a single logarithm.
Exercise #22
This document outlines three theorems for logarithms:
1) The Product Theorem states that the log of a product is equal to the sum of the logs of the factors.
2) The Quotient Theorem states that the log of a quotient is equal to the difference of the logs of the factors.
3) The Power Theorem states that the log of a factor to a given power is equal to the power times the log of the factor.
These theorems only apply when the logarithms have the same base. Examples are provided to demonstrate applying the theorems to simplify logarithmic expressions and evaluate logarithms.
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This document discusses the derivative of logarithmic functions and provides an example of taking the derivative of logb(x) with respect to x. It also gives an example of a matrix with elements containing fractions.
Log summary & equations
Here are solutions to the practice problems:
log(2x – 4) = 3
2x – 4 = 23 = 8
2x – 4 = 8
2x = 12
x = 6
log2(x) + log2 (x – 6) = 4
log2(x(x – 6)) = 4
x(x – 6) = 24
x2 – 6x – 24 = 0
(x – 6)(x + 4) = 0
x = 6, x = -4
log4(x + 4) – log4(x – 1) = 2
log4((x + 4)/(x – 1)) = 2
(x
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.
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.
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This document discusses the properties and laws of logarithms. It explains that common and natural logarithms have the same properties and laws, even though their bases differ. The key properties covered include: logarithms are only defined for positive values; ln1=0, ln e=1; and the product, quotient, and power laws. Several examples are worked through to demonstrate how to use these properties and laws to simplify logarithmic expressions and solve equations. The document emphasizes that understanding logarithmic properties and how to manipulate logarithms is important for advanced math, science, and engineering courses.
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The document discusses binomial expansion, which is the process of multiplying out terms with two variables according to their power using the binomial theorem. It provides examples of expanding binomial expressions like (x + y)2, (x + y)3, and (x + y)4. It also notes that the sum of the exponents in each term equals the overall power, and the number of terms is always one more than the power. Finally, it provides the binomial theorem for expanding any binomial expression and finding a particular term.
Day 4 examples u7w14
This document discusses how to calculate arrangements when some items must be together or apart. It explains that when items need to be together, they should be counted as a single item to reduce the total items being arranged. Then the total number of arrangements is calculated by finding the total possible arrangements and subtracting the arrangements that do not satisfy the constraints of certain items being together or apart. Examples provided include arranging people in a row when some must or cannot sit together and arranging books on a shelf keeping books of each subject together.
Day 3 examples u7w14
Permutations refer to arrangements of objects in a definite order. Some key points:
- Permutations are represented by "nPn" where n is the total number of objects and r is the number being arranged.
- Permutations are used to calculate possibilities like license plates, phone numbers, and locker combinations.
- Restrictions like starting/ending conditions or requiring alternating arrangements reduce the number of possible permutations.
- Objects that are identical only count once toward the total number of permutations rather than being distinguishable.
Day 2 examples u7w14
Factorial notation represents the product of all positive integers less than or equal to the given number. For example, 5! = 5 x 4 x 3 x 2 x 1 and 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. The document also provides examples of simplifying factorials without a calculator by using properties such as 5! + 4! = 6 x 4! and (k + 1)! + k! = (k + 2)k!.
Day 1 examples u7w14
This document provides examples and explanations of the fundamental counting principle and addition counting principle to solve combinatorics problems. It gives 8 examples of using the fundamental counting principle to count the number of possible outcomes of independent events. These include counting the number of volleyball shoe combinations, outfits that can be created from different clothing items, ways to select committees from groups of people, and 3-digit numbers with no repeating digits. It also provides 5 examples of using the addition counting principle to count outcomes when events are dependent, such as selecting a president and vice president of opposite sexes from a group.
Day 7 examples u6w14
This document discusses graphing composite functions. It provides examples of determining the composite functions f(g(x)) and g(f(x)) for various functions f(x) and g(x), sketching the graphs of the composite functions, and stating their domains. It also gives examples of determining possible functions f(x) and g(x) that satisfy given composite functions.
Day 5 examples u6w14
1. The document discusses composite functions, which involve combining two functions f(x) and g(x) where the output of one is used as the input of the other. It provides examples of evaluating composite functions using tables and graphs.
2. Key steps for evaluating composite functions are: 1) Substitute the inner function into the outer function and 2) Simplify the expression. Order matters as f(g(x)) and g(f(x)) may have different values.
3. Examples are worked through to find composite functions given basic functions like f(x) = x + 1 and g(x) = 2x as well as more complex rational functions.
Day 4 examples u6w14
The graph is a linear function with a domain of all real numbers and a range of real numbers greater than or equal to 3. The graph is a line with a y-intercept of 3 that increases at a rate of 1 as x increases.
Day 3 examples u6w14
Rational functions are functions of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomials. For example, comparing rational functions like 2x/(x^2 - 4) and (x-1)/(x+1). Horizontal asymptotes of rational functions occur when the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
Day 2 examples u6w14
This document discusses combining functions by graphing. When two functions f(x) and g(x) are combined, their graphs are overlayed on the same coordinate plane. The result is a new combined function where the output is determined by applying both functions f(x) and g(x) to the same input x.
Day 1 examples u6w14
This document discusses how to find the sum, difference, product, and quotient of functions. The sum of functions is found by adding the y-coordinates of each function. The difference is found by subtracting the y-coordinates. The product is represented as h(x) = f(x)g(x) and the quotient is represented as h(x) = f(x)/g(x). Examples are provided for adding and subtracting functions.
Mental math test outline
The document outlines a mental math test covering polynomials. It includes short answer questions testing long division, synthetic division, the remainder theorem, and finding the degree, leading coefficient, and y-intercept of polynomials. The test also covers matching graphs to polynomial equations and word problems involving fully factoring polynomials and two graphs. Multiple choice questions will require explaining solutions, while long answer questions involve fully factoring polynomials and word problems.
Day 8 examples u5w14
The document contains two polynomial word problems. The first asks to write a function V(x) to express the volume of a box with dimensions x, x+2, x+10 in terms of x, and find possible x values if the volume is 96 cm^3. The second problem describes a block of ice that is initially 3 ft by 4 ft by 5 ft, and asks to write a function to model reducing each dimension by the same amount to reach a volume of 24 ft^3, and determine how much to remove from each dimension.
Day 7 examples u5w14
The document provides 3 polynomial word problems: 1) finding the equation for a polynomial given its graph f(x) = -(x - 2)2(x + 1), 2) determining the polynomial P(x) when divided by (x - 3) with a quotient of 2x^2 + x - 6 and remainder of 4, and 3) finding the value of a if (x - 2) is a factor of ax^3 + 4x^2 + x - 2. It also gives a 4th problem of determining the value of k when 2x^3 + kx^2 - 3x + 2 is divided by x - 2 with a remainder of 4.
Day 5 examples u5w14
Polynomial functions are described by their degree and have certain characteristics. The graph of a polynomial is smooth and continuous without sharp corners. Odd degree polynomials rise on the left and fall on the right, while even degree polynomials rise on both sides. The number of x-intercepts and local maxima/minima are limited by the degree. Polynomials can be matched based on their degree, leading coefficient, even/odd nature, and number of x-intercepts and local extrema. The x-intercepts of a polynomial correspond to the roots of the equation, and a repeated root indicates a zero of higher multiplicity which affects the graph.
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New day 3 examples
- 1. Log Properties/ Theorems/ Laws single log expanded log (M · N) = log Extension: log 8 = log 8
- 3. Use Log Laws to Evaluate Expressions ex 1) log6 8 + log6 9 - log6 2 ex 2) log7 7√7 1 ex 3) 2log2 12 - (log2 6 + 3 log2 27)
- 4. Rewrite as a single log: (x - 2) - 3log alog3 - logx - log(a + 1) ex 3) log7 x2 + log7 x - 5log7 x 2