Logarithmic Functions Digital Lesson Logarithmic Functions

Definition: Logarithmic Function For x  0 and 0  a  1, y = loga x if and only if x = a y. The function given by f (x) = loga x is called the logarithmic function with base a. Every logarithmic equation has an equivalent exponential form: y = loga x is equivalent to x = a y A logarithm is an exponent! A logarithmic function is the inverse function of an exponential function. Exponential function: y = ax Logarithmic function: y = logax is equivalent to x = ay Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Logarithmic Function

Examples: Write Equivalent Equations Examples: Write the equivalent exponential equation and solve for y. Solution Equivalent Exponential Equation Logarithmic Equation y = log216 16 = 2y 16 = 24  y = 4 y = log2( ) = 2 y = 2-1 y = –1 y = log416 16 = 4y 16 = 42  y = 2 y = log51 1 = 5 y 1 = 50  y = 0 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Examples: Write Equivalent Equations

Common Logarithmic Function The base 10 logarithm function f (x) = log10 x is called the common logarithm function. The LOG key on a calculator is used to obtain common logarithms. Examples: Calculate the values using a calculator. Function Value Keystrokes Display log10 100 LOG 100 ENTER 2 log10( ) LOG ( 2 5 ) ENTER – 0.3979400 log10 5 LOG 5 ENTER 0.6989700 log10 –4 LOG –4 ENTER ERROR no power of 10 gives a negative number Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Common Logarithmic Function

Properties of Logarithms 1. loga 1 = 0 since a0 = 1. 2. loga a = 1 since a1 = a. 3. loga ax = x and alogax = x inverse property 4. If loga x = loga y, then x = y. one-to-one property Examples: Solve for x: log6 6 = x log6 6 = 1 property 2 x = 1 Simplify: log3 35 log3 35 = 5 property 3 Simplify: 7log79 7log79 = 9 property 3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Properties of Logarithms

Example: Graph the common logarithm function f(x) = log10 x. 0.602 0.301 –1 –2 f(x) = log10 x 10 4 2 x y x 5 –5 by calculator f(x) = log10 x (0, 1) x-intercept x = 0 vertical asymptote Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: f(x) = log0 x

Graphs of Logarithmic Functions The graphs of logarithmic functions are similar for different values of a. f(x) = loga x (a  1) y-axis vertical asymptote x y Graph of f (x) = loga x (a  1) y = a x y = x range 1. domain y = log2 x 2. range 3. x-intercept (1, 0) 4. vertical asymptote domain x-intercept (1, 0) 5. increasing 6. continuous 7. one-to-one 8. reflection of y = a x in y = x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Graphs of Logarithmic Functions

Natural Logarithmic Function y x 5 –5 y = ln x The function defined by f(x) = loge x = ln x (x  0, e 2.718281) is called the natural logarithm function. y = ln x is equivalent to e y = x Use a calculator to evaluate: ln 3, ln –2, ln 100 Function Value Keystrokes Display ln 3 LN 3 ENTER 1.0986122 ln –2 LN –2 ENTER ERROR ln 100 LN 100 ENTER 4.6051701 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Natural Logarithmic Function

Properties of Natural Logarithms 1. ln 1 = 0 since e0 = 1. 2. ln e = 1 since e1 = e. 3. ln ex = x and eln x = x inverse property 4. If ln x = ln y, then x = y. one-to-one property Examples: Simplify each expression. inverse property inverse property property 2 property 1 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Properties of Natural Logarithms