 # Chapter 4 Exponential and Logarithmic Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 4.2 Logarithmic Functions.

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Chapter 4 Exponential and Logarithmic Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 4.2 Logarithmic Functions

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Definition of the Logarithmic Function For x > 0 and b > 0, b ≠1, y = log b x is equivalent to b y = x. The function f(x) = log b x is the logarithmic function with base b.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Example: Changing from Logarithmic to Exponential Form Write each equation in its equivalent exponential form:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Example: Changing from Exponential to Logarithmic Form Write each equation in its equivalent logarithmic form:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Evaluating Logarithms Evaluate:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Basic Logarithmic Properties Involving One 1. log b b = 1 because 1 is the exponent to which b must be raised to obtain b. (b 1 = b) 2. log b 1 = 0 because 0 is the exponent to which b must be raised to obtain 1. (b 0 = 1)

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Using Properties of Logarithms Evaluate: log 9 9 log 8 1

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Inverse Properties of Logarithms For b > 0 and b ≠ 1,

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Using Inverse Properties of Logarithms Evaluate:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Graphs of Exponential and Logarithmic Functions (continued) We are graphing and x –2–2 –1–1 0 1 x – 2 –1 10 31

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Characteristics of Logarithmic functions of the Form

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 The Domain of a Logarithmic Function The domain of an exponential function of the form includes all real numbers and its range is the set of positive real numbers. Because they are inverses, logarithmic function reverses the domain and the range of the exponential function.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Finding the Domain of a Logarithmic Function Find the domain of

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Common Logarithms The logarithmic function with base 10 is called the common logarithmic function. The function is usually expressed

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Application The percentage of adult height attained by a boy who is x years old can be modeled by where x represents the boy’s age (from 5 to 15) and f(x) represents the percentage of his adult height. Approximately what percentage of his adult height has a boy attained at age ten?

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Properties of Common Logarithms

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Natural Logarithms The logarithmic function with base e is called the natural logarithmic function. The function is usually expressed

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Properties of Natural Logarithms

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Application When the outside air temperature is anywhere from 72° to 96° Fahrenheit, the temperature in an enclosed vehicle climbs by 43° in the first hour. The function models the temperature increase, f(x), in degrees Fahrenheit, after x minutes. Use the function to find the temperature increase, to the nearest degree, after 30 minutes.