C ollege A lgebra

1. College Algebra Inverse Functions ; Exponential and Logarithmic Functions(Chapter4) 1

2. sections 4.5,4.6,4.7& EquationsLogarithmic Functions Objectives: • After completing this tutorial, you should be able to: • Know the definition of a logarithmic function.  • Write a log function as an exponential function and vice versa.  • Graph a log function.  • Evaluate a log.  • Be familiar with and use properties of logarithms in various situations. • Solve logarithmic equations.  

3. Definition of Log Function • For all real numbers y, and all positive numbers a (a > 0) and x, where a 1: • Meaning of logax A logarithm is an exponent; logax is the exponent to which the base a must be raised to obtain x. (Note: Logarithms can be found for positive numbers only) A LOG IS ANOTHER WAY TO WRITE AN EXPONENT. 

4. Exponent Exponent Base Base Location of Base and Exponent in Exponential and Logarithmic Forms Logarithmic form: y = logb x Exponential Form: by = x.

5. Example :  Express the logarithmic equation exponentially

6. a. 2 = log5x means 52 = x. b. 3 = logb 64 means b3 = 64. Logarithms are exponents. Logarithms are exponents. Logarithms are exponents. Logarithms are exponents. Examples Write each equation in its equivalent exponential form. a. 2 = log5x b. 3 = logb 64 c. log3 7 = y Solution With the fact that y = logbx means by = x, c. log3 7 = y or y = log3 7 means 3y = 7.

7. Evaluating Logs

8. Evaluating Logs Example :  Evaluate the expression without using a calculator.

9. Logarithmic Expression Question Needed for Evaluation Logarithmic Expression Evaluated a. log2 16 2 to what power is 16? log2 16 = 4 because 24 = 16. b. log3 9 3 to what power is 9? log3 9 = 2 because 32 = 9. c. log25 5 25 to what power is 5? log25 5 = 1/2 because 251/2 = 5. Text Example Evaluate a. log2 16 b. log3 9 c. log25 5 Solution

10. Graphing Log Functions

11. Graphing Log Functions

12. Characteristics of the Graph of f(x) = logax • The points (1, 0), and (a, 1) are on the graph. • If a > 1, then f is an increasing function; if 0 < a < 1, then f is a decreasing function. • The y-axis is a vertical asymptote. • The domain is (0, ), and the range is (, ).

13. Graph Write in exponential form as Now find some ordered pairs. x y 1 0 1/16 2 4 1 Example

14. Graph Write in exponential form as Now find some ordered pairs. x y 1 0 5 1 0.2 1 Example

15. Graph the function. The vertical asymptote is x = 1. To find some ordered pairs, use the equivalent exponent form. Translated Logarithmic Functions

16. Graph To find some ordered pairs, use the equivalent exponent form. Translated Logarithmic Functions continued

17. Property Description Product Property The logarithm of a product of two numbers is equal to the sum of the logarithms of the numbers Quotient Property The logarithm of the quotient of two numbers is equal to the difference between the logarithms of the numbers. Power Property The logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number. Properties of Logarithms, For x > 0, y > 0, a > 0, a 1, and any real number r:

18. Using the Properties of Logarithms • Rewrite each expression. Assume all variables represent positive real numbers with a 1 and b  1. • a) • b) • c)

19. Using the Properties of Logarithms • Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with a 1 and b  1. • a) • b)

20. Using the Properties of Logarithms Expand as much as possible. Evaluate without a calculator where possible

21. Inverse Properties of Logarithms Inverse Property I • For a > 0, a 1: • By the results of this theorem:

22. Inverse Properties of Logarithms Inverse Property II • For b > 0, b 1: • By the results of this theorem: b logb x = x , 

23. Basic Logarithmic Properties Involving One Logbb = 1 because 1 is the exponent to which b must be raised to obtain b. (b1 = b). Logb 1 = 0 because 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).

24. General PropertiesCommon Logarithms 1. logb 1 = 0 1. log 1 = 0 2. logbb = 1 2. log 10 = 1 3. logbbx = 0 3. log 10x = x 4. b logb x = x 4. 10 log x = x Properties of Common Logarithms Examples of Logarithmic Properties log b b = 1 log b 1 = 0 log 4 4 = 1 log 8 1 = 0 3 log 3 6 = 6 log 5 5 3 = 3 2 log 2 7 = 7

25. Natural Logarithms • Logarithms with a base of e are referred to a natural logarithms. • So if f(x) = ex , then f(x) = loge x = lnx • Recall, e = 2.71828 Properties of Natural Logarithms General Natural Properties Logarithms 1. logb 1 = 0 1. ln 1 = 0 2. logbb = 1 2. ln e = 1 3. logbbx = 0 3. ln ex = x 4. b logb x = x 4. eln x = x Examples log e e = 1 log e 1 = 0 e log e 6 = 6 log e e 3 = 3

26. Change-of-Base Theorem • For any positive real numbers x, a, and b, where a 1 and b  1: • logax =lnx/ lna

27. a) log512 b) log2.4 Examples Use the change-of-base theorem to find an approximation to four decimal places for each logarithm.

28. Solving Exponential or Logarithmic Equations

29. Solving Logarithmic Equations • Solve each equation. • a) b)

30. Example • Solve8x= 15 • The solution set is {1.3023}.

31. Example • Solvecontinued

32. Example • Solve

33. Example • Solve The only valid solution is x = 4.

34. Example • Solve

35. Example • Solvecontinued The only valid solution is x = 2.

36. Notes, review

37. Properties of Logarithms

38. Write the following expression as the sum and/or difference of logarithms. Express all powers as factors.

39. Write the following expression as a single logarithm.

40. Most calculators only evaluate logarithmic functions with base 10 or base e. To evaluate logs with other bases, we use the change of base formula.

41. Practices

45. No Solution

47. End of the Lecture Let Learning Continue