Introduction to Logarithmic Functions Unit 3: Exponential and Logarithmic Functions

Introduction to Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE 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. f(x) y = 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 = 2x Table of values:

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

Introduction to Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE When we add the function f(x) = 2x to this graph, it is evident that the inverse is a reflection on the y = x axis f(x) f-1(x) 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 exponent y = ax x = ay Interchange x  y x = ay We write these functions as: x = ay y = logax y = logax base exponent

Introduction to Logarithmic Functions FINDING THE INVERSE OF AN EXPONENTIAL x y y x log = a Inverse of the Exponential Function Logarithmic 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 log327=3 b) 45 = 256 log4256=5 c) 27 = 128 log2128=7 d) (1/3)x=27 log1/327=x

Introduction to Logarithmic Functions CHANGING FORMS Example 2) Write the following into exponential form: a) log264=6 b) log255=1/2 c) log81=0 d) log1/39=2 ANSWERS

Introduction to Logarithmic Functions CHANGING FORMS Example 2) Write the following into exponential form: a) log264=6 26 = 64 b) log255=1/2 251/2 = 5 c) log81=0 80 = 1 (1/3)2 = 1/9 d) log1/39=2

Introduction to Logarithmic Functions EVALUATING LOGARITHMS Example 3) Find the value of x for each example: a) log1/327 = x b) log5x = 3 c) logx(1/9) = 2 d) log3x = 0 ANSWERS

Introduction to Logarithmic Functions EVALUATING LOGARITHMS Example 3) Find the value of x for each example: a) log1/327 = x b) log5x = 3 (1/3)x = 27 (1/3)x = (1/3)-3 x = -3 53 = x x = 125 c) logx(1/9) = 2 d) log3x = 0 x2 = (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 log10 Example 4) Use your calculator to find the value of each of the following: a) log101000 b) log 50 c) log -1000 = Out of Domain = 3 = 1.699

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