Introduction to Logarithmic Functions Unit 3 Exponential and

Introduction to Logarithmic Functions Unit 3: Exponential and Logarithmic Functions

Introduction to Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE l. In Grade 11, you were introduced to inverse functions. • Inverse functions is the set of ordered pair obtained by interchanging the x and y values. f(x) f-1(x)

Introduction to Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE • Inverse functions can be created graphically by a reflection on the y = x axis. y = x f(x) f-1(x)

Introduction to Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE • A logarithmic function is the inverse of an exponential function • Exponential functions have the following characteristics: Domain: {x є R} Range: {y > 0}

Introduction to Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE • Let us graph the exponential function y = 2 x • Table of values:

Introduction to Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE • Let us find the inverse the exponential function y = 2 x • Table of values:

Introduction to Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE • When we add the function f(x) = 2 x to this graph, it is evident that the inverse is a reflection on the y = x axis f(x) f-1(x)

Introduction to Logarithmic Functions FINDING THE INVERSE OF AN EXPONENTIAL • Next, you will find the inverse of an exponential algebraically • Write the process in your notes base y = ax Interchange x y x = ay • We write these functions as: x = ay exponent y = logax exponent base

Introduction to Logarithmic Functions FINDING THE INVERSE OF AN EXPONENTIAL x y a y x =log Logarithmic Inverse of the Form Exponential Function

Introduction to Logarithmic Functions CHANGING FORMS Example 1) Write the following into logarithmic form: a) 33 = 27 b) 45 = 256 c) 27 = 128 d) (1/3)x=27 ANSWERS

Introduction to Logarithmic Functions CHANGING FORMS Example 1) Write the following into logarithmic form: a) 33 = 27 log 327=3 b) 45 = 256 log 4256=5 c) 27 = 128 log 2128=7 d) (1/3)x=27 log 1/327=x

Introduction to Logarithmic Functions CHANGING FORMS Example 2) Write the following into exponential form: a) log 264=6 b) log 255=1/2 c) log 81=0 d) log 1/39=-2 ANSWERS

Introduction to Logarithmic Functions CHANGING FORMS Example 2) Write the following into exponential form: a) log 264=6 26 = 64 b) log 255=1/2 251/2 = 5 c) log 81=0 80 = 1 d) log 1/39=-2 (1/3)-2 = 9

Introduction to Logarithmic Functions EVALUATING LOGARITHMS Example 3) Find the value of x for each example: a) log 1/327 = x b) log 5 x = 3 c) logx(1/9) = 2 d) log 3 x = 0 ANSWERS

Introduction to Logarithmic Functions EVALUATING LOGARITHMS Example 3) Find the value of x for each example: a) log 1/327 = x (1/3)x = 27 (1/3)x = (1/3)-3 x = -3 b) log 5 x = 3 53 = x x = 125 c) logx(1/9) = 2 d) log 3 x = 0 x 2 = (1/9) x = 1/3 30 = x x=1

Introduction to Logarithmic Functions BASE 10 LOGS Scientific calculators can perform logarithmic operations. Your calculator has a LOG button. This button represents logarithms in BASE 10 or log 10 Example 4) Use your calculator to find the value of each of the following: a) log 101000 b) log 50 c) log -1000 = Out of Domain =3 = 1. 699

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