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

horizontal asymptote y = 0 Graph f (x) = log2 x Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x. x y y = 2x y = x 8 3 4 2 1 –1 –2 2x x horizontal asymptote y = 0 y = log2 x x-intercept (1, 0) vertical asymptote x = 0 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Graph f(x) = log2 x

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

Example: Carbon Dating Example: The formula (t in years) is used to estimate the age of organic material. The ratio of carbon 14 to carbon 12 in a piece of charcoal found at an archaeological dig is . How old is it? original equation multiply both sides by 1012 take the natural log of both sides inverse property To the nearest thousand years the charcoal is 57,000 years old. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Carbon Dating