1. Inverse Functions ; Exponential and Logarithmic Functions (Chapter4)
University of Palestine
IT-College
College Algebra
Inverse Functions ; Exponential and Logarithmic Functions (Chapter4)
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2. Definition and Graph of the Natural Exponential Function

4. The Natural Base e
An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to More accurately,
The number e is called the natural base. The function f (x) = ex is called the natural exponential function. -1
f (x) = ex
f (x) = 2x
f (x) = 3x
(0, 1)
(1, 2) 1 2 3 4
(1, e)
(1, 3)

5. Solving Exponential Equations,  where x is in the exponent, BUT the bases DO NOT MATCH.
Step 1: Isolate the exponential expression.
Get your exponential expression on one side everything outside of the exponential expression on the other side of your equation.
Step 2: Take the natural log of both sides.
The inverse operation of an exponential expression is a log.  Make sure that you do the same thing to both sides of your equation to keep them equal to each other.
Step 3: Use the properties of logs to pull the x out of the exponent.
Step 4: Solve for x.
Now that the variable is out of the exponent, solve for the variable using inverse operations to complete the problem.

6. Example 1: Solve the exponential equation
Round your answer to two decimal places.
Step 1: Isolate the exponential expression.
This is already done for us in this problem.
Step 2: Take the natural log of both sides.
Step 3: Use the properties of logs to pull the x out of the exponent.

7. Step 4: Solve for x.

8. Example 2: Solve the exponential equation
Round your answer to two decimal places.
Step 1: Isolate the exponential expression.
Step 2: Take the natural log of both sides.
Step 3: Use the properties of logs to pull the x out of the exponent.

9. Step 4: Solve for x.

10. Example 3: Solve the exponential equation
Round your answer to two decimal places.
Step 1: Isolate the exponential expression.
            
Step 2: Take the natural log of both sides.
Step 3: Use the properties of logs to pull the x out of the exponent.

11. Step 4: Solve for x.

12. Exponential Equations
Solve

13. Another Example
Solve 3x + 1 = 27x  3

14. sections 4.5,4.6,4.7 & Equations Logarithmic 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.  

15. 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. 

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

17. Example : Express the logarithmic equation exponentially
We want to use the definition that is above:                 if and only if        .
                          

18. Examples Write each equation in its equivalent exponential form.
a. 2 = log5 x b. 3 = logb c. log3 7 = y
Solution With the fact that y = logb x means by = x,
a. 2 = log5 x means 52 = x.
Logarithms are exponents.
b. 3 = logb 64 means b3 = 64.
Logarithms are exponents.
c. log3 7 = y or y = log3 7 means 3y = 7.

19. Evaluating Logs Step 1: Set the log equal to x.
Step 2:  Use the definition of logs shown above to write the equation in exponential form. 
Step 3: Find x.
Whenever you are finding a log, keep in mind that logs are another way to write exponents.  You can always use the definition to help you evaluate. 

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

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

22. Graphing Log Functions

23. Graphing Log Functions

24. 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 (, ).

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

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

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

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

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

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

31. 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)

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

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

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

35. Basic Logarithmic Properties Involving One
Logb b = 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).

36. Properties of Common Logarithms
General Properties Common Logarithms
1. logb 1 = log 1 = 0
2. logb b = log 10 = 1
3. logb bx = x log 10x = x
4. b logb x = x log x = x
Examples of Logarithmic Properties
log b b = log b 1 = log 4 4 = 1
log 8 1 = log 3 6 = log = 3
2 log 2 7 = 7

37. Properties of 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 =
Properties of Natural Logarithms
General Natural
Properties Logarithms
1. logb 1 = ln 1 = 0
2. logb b = ln e = 1
3. logb bx = ln ex = x
4. b logb x = x 4. e ln x = x
Examples
log e e = 1
log e 1 = 0
e log e 6 = 6
log e e 3 = 3

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

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

40. Solving Exponential or Logarithmic Equations

41. Solving Logarithmic Equations
Solve each equation.
a) b)

42. Example
Solve 8x = 15
The solution set is {1.3023}.

43. Example
Solve continued

44. Example
Solve

45. Example
Solve
The only valid solution is x = 4.

46. Example
Solve

47. The only valid solution is x = 2.
Example
Solve continued
The only valid solution is x = 2.

48. Notes, review

49. Properties of Logarithms

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

51. Write the following expression as a single logarithm.

52. 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.

53. Practices

57. No Solution

59. End of the Lecture
Let Learning Continue