1. Logarithmic Functions
Algebra III, Sec. 3.2
Objective
Recognize, evaluate, and graph logarithmic functions.

2. Important Vocabulary
Common logarithmic function – logarithm with base 10
Natural logarithmic function – logarithm with base e

3. Logarithmic Functions
The logarithmic function with base a is defined as ___________________ for x > 0 and 0 < a ≠ 1, if and only if x = ay. The logarithmic function with base a is the ________________ of the exponential function f(x) = ax.
inverse

4. Logarithmic Functions (cont.)
The equation x = ay in exponential form is equivalent to the equation _______________ in logarithmic form. When evaluating logarithms, remember that a logarithm is an ________________. This means that logax is the _______________ to which a must be raised to obtain ______.
exponent
exponent x

5. Example (on your handout)
Use the definition of logarithmic function to evaluate

6. Example 1
Evaluate each logarithm at the indicated value of x. a.) at x = 16 b.) at x = 64 c.) at x = 1 d.) at x = 1/81 2 6 -4

7. Example (on your handout)
Use a calculator to evaluate
2.4771

8. Example 2
Use a calculator to evaluate the function f(x) = log x at each value of x. a.) x = 100 b.) x = 1/5 c.) x = 3.25 d.) x = -4 2
0.5119
error

9. Properties of Logarithms

10. Example 3
Simplify a.) b.) c.) 1 30

11. Example (on your handout)
Solve the equation for x.

12. Example 4
Solve a.) b.) c.)
y = 16
x = ½
x = ±5

13. Graphs of Logarithmic Functions
For a > 1, the graph of y = logax is _________________ over its domain. For the graph of y = logax, a > 1, the domain is ________________, the range is ________________, and the x-intercept is _________________.
increasing
(0, ∞)
(-∞, ∞)
(1, 0)

14. Graphs of Logarithmic Functions
Also, the graph has ________________ as a vertical asymptote. The graph of y = logax is a reflection of the graph of y = ax about _______________.
the y-axis
y = x

15. Example 5
In the same coordinate plane, sketch the graph of each function. a.) b.)

16. Example 6
Sketch the graph of the function

17. Example (on your handout)
Sketch the graph of the function

18. Example 7
Describe the graph as a transformation of the graph of a.) b.)

19. The Natural Logarithmic Function

20. Example 8
Use a calculator to evaluate the function at each value of x. a.) x = b.) x = 0.4 c.) x = -2 d.) x = 2 + √3
5.2939
0.0837
Error
2.3170

21. Example (on your handout)
Use a calculator to evaluate
2.3026

22. Example 9
Use the properties of natural logarithms to simplify each expression. a.) b.) c.) d.) 8
1/6

23. Example (on your handout)
Find the domain of the function

24. Application