Chapter 12 Exponential and Logarithmic Functions What is a logarithmic function?

Inverse Relations and Functions 12-1 What is the inverse of a function?

ACTIVATION: Given 3 What is its additive inverse: What is its multiplicative inverse: What do you think of when you hear the term inverse?

Inverse of a Function Inverse relation—interchanging the x and y Inverse function—the inverse of a relation has one y for each and every x Given G={(1, 3), (2, 4), (6, 3), (7, 7)} Find the inverse = {(3, 1), (4, 2), (3, 6), (7,7)}

Finding the inverse of an equation: Find the inverse of y = 4x – 5 What is there to change? x = 4y – 5 x + 5 = 4y x + 5 = y-1 4

Examples: y = 5x2 – 4

Examples: y-1 = x2 -1 For x > 0 Why is the domain of the inverse restricted? The domain of the original must be x+1 >0 or x>-1 Since the x and y change for the inverse y>-1 How can this happen? y-1 = x2 -1 For x > 0

Test for symmetry to y = x If you interchange the x and y and get the original equation the function is symmetric to the line y = x 3x + 3y = 8 4x – 4y = 3 Switch 3y + 3x = 8 4y – 4x = 3 addition is commutative subtraction is not has symmetry to y = x does not have symmetry to y = x

Proving 2 functions are inverses Means they will be reflected over y = x or If we find f(x) and then test that in f-1(x) we should arrive back at the original number f(f-1(x)) =x AND f-1(f(x)) = x Given f(x) = x3 – 5 and g(x) = are the functions inverses Test: f(g(x)) and g(f(x))

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Exponential and Logarithmic Functions 12-2 What are exponential and logarithmic functions?

ACTIVATION: Graph y = 2x x = 2y What is true of these two functions? x the graphs show they are inverse but graphing the second is awkward x y -2 -1 1 2 3 y .25 .5 1 2 4 8 x y-1 -2 -1 1 2 3 x .25 .5 1 2 4 8

The log is the inverse of the exponential function Use your calculator to graph y = 10x and y = log x

The transformation equation: y = a func( x – h) + k Will it work with the exponential and log equations Determines Up/Down movement +/ - Determines Width and if it is inverted Determines Left/Right movement +/ -

Transforming exponential and log functions work with a partner to sketch the graphs

Transforming exponential and log functions work with a partner to sketch the graphs what about y = (1/2)x y = (2-1)x y = 2-x

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Exponential and Logarithmic Functions 12-3 Day 3 Exponential and Logarithmic Functions 12-3 How are exponential and logarithmic functions related? Properties of Logarithmic Functions 12-4 What are the basic properties of logarithmic functions? Exponential and Logarithmic Equations 12-5 How do you solve exponential and logarithmic equations?

ACTIVATION: Simplify: x2x4 x-3 Since exponential and logarithmic equations are inverses there are some properties which are similar. Such as multiplying and addition are related: as are division and subtraction: and raising a power to a power means to multiply

Primary Rule of Logarithms Primary rule of logs: logb x = y becomes x = by allows us to convert from the exponential equation to a logarithmic equation or the other way Try: log 3 12 = x and x = 43 12 = 3x log4 x = 3

What would be true of the following and WHY???? loga a = 1 loga x = 0 NOTE: the log of anything to its own base is 1 when the solution is 0 you are taking the log of 1

Examples: Solve log5 125 = x logx 81 = 4 log6 216 = x

Examples: Solve log3 x = -3 log4 = x log 1000 = x

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Exponential and Logarithmic Functions 12-3 Day 4 Exponential and Logarithmic Functions 12-3 How are exponential and logarithmic functions related? Properties of Logarithmic Functions 12-4 What are the basic properties of logarithmic functions? Exponential and Logarithmic Equations 12-5 How do you solve exponential and logarithmic equations?

How are the laws for exponents and logarithms related Let m = loga x x = am Written in exponent form xn = ( am )n Raise both sides to the power of n log a xn = mn Convert back to a logarithmic equation log a xn = n loga x Substitute for m When a “power” is raised to a power multiply them.

By the same type of proofs let b = logax and c = logay convert x=ab y = ac multiply xy =abac xy = ab+c take the log of both loga xy =loga ab+c convert loga xy = b + c substitute loga xy = logax + loga y

By the same type of proofs

Examples: log 800 + log 5 – log 40

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Exponential and Logarithmic Functions 12-3 Day 4 Exponential and Logarithmic Functions 12-3 How are exponential and logarithmic functions related? Properties of Logarithmic Functions 12-4 What are the basic properties of logarithmic functions? Exponential and Logarithmic Equations 12-5 How do you solve exponential and logarithmic equations?

Examples log (x2 -1) – log (x+2) = 1 log (2x -15) log x =2

Example: log (3x + 2) + log (4x – 1) = 2 log 11

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Lesson 12-8 What are the natural log and the number e? What is the change of base ?

ACTIVATION: What is THE primary rule of equations? —whatever you do to one side you must do to the other. Given 3x = 5 solve for x

Change of base theorem: used when the base and the number are not powers of the same value. logb x = y x = by log x = log by log x = y log b log x = y log b

Examples: Solve log3 12 = x log5 25 = x

Applications: Exponential growth Exponential growth follows the model P = P0 ert P = final amount P0 = initial amount r = rate as a decimal t = time

Applications: Exponential growth Exponential growth follows the model P = P0 ert The population of the US was about 203 million in 1970. In 1989, it was about 247 million. Find the growth rate of the population. 247 = 203 er(1989-1970)

Applications: Exponential growth A particular radioactive isotope has a half-life of 2 years. A scientist has 224 grams on hand. How much of the substance will remain as a radio active isotope after 24 years. ½ P0 = P0 e-r • 2 (this is a two step process) ln ½ = ln e-2r -.693 = -2r .346 = r then P = 224 e-(.346) (24) P = .054 or A=P(1 + )nt A = 224 (1 + )1•12 A = .054

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