Lecture 11 sections 4.3-4.4 logarithmic functions

1. MATH 108 Section 4.3 Logarithmic Functions

2. 4 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Converting from Exponential to Logarithmic Form Write each exponential equation in logarithmic form. Solution

3. 6 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Converting from Logarithmic Form to Exponential Form Write each logarithmic equation in exponential form. Solution

4. (a) 3 raised to what power yields 81? (b) 2 raised to what power yields ?

5. 9 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Evaluating Logarithms Find the value of each of the following logarithms. Solution

6. 10 © 2010 Pearson Education, Inc. All rights reserved COMMON LOGARITHMS log 10 = 1 log 1 = 0 log 10 x = x The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: log x = log 10 x. Thus, y = log x if and only if x = 10 y . Applying the basic properties of logarithms

7. 11 © 2010 Pearson Education, Inc. All rights reserved NATURAL LOGARITHMS ln e = 1 ln 1 = 0 log e x = x The logarithm with base e is called the natural logarithm and is denoted by ln x. That is, ln x = log e x. Thus, y = ln x if and only if x = e y . Applying the basic properties of logarithms

8. Natural Logarithm Function

9. 15 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Using Transformations Start with the graph of f ( x ) = log 3 x and use transformations to sketch the graph of each function. State the domain and range and the vertical asymptote for the graph of each function.

10. a. b. c.

11. MATH 108 Section 4.4 Properties of Logarithms

12. 20 © 2010 Pearson Education, Inc. All rights reserved RULES OF LOGARITHMS Let M , N , and a be positive real numbers with a ≠ 1, and let r be any real number. The logarithm of the product of two (or more) numbers is the sum of the logarithms of the numbers. 1. Product Rule

13. 21 © 2010 Pearson Education, Inc. All rights reserved RULES OF LOGARITHMS Let M , N , and a be positive real numbers with a ≠ 1, and let r be any real number. The logarithm of the quotient of two (or more) numbers is the difference of the logarithms of the numbers. 2. Quotient Rule

14. 22 © 2010 Pearson Education, Inc. All rights reserved RULES OF LOGARITHMS Let M , N , and a be positive real numbers with a ≠ 1, and let r be any real number. The logarithm of a number to the power r is r times the logarithm of the number. 3. Power Rule

15. 27 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Writing Expressions In Expanded Form Write each expression in expanded form. Solution

16. 29 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Writing Expressions in Condensed Form Write each expression in condensed form.