1. Logarithmic Functions
Section 11.3
Logarithmic Functions

2. Definition of a Logarithm
For b > 0, b ≠ 1, and a > 0, the logarithm logb (a) is the number k such that bk = a. We call b the base of the logarithm.
Rewrite each logarithmic expression as an exponential and simplify.
Example

3. Definition of a Logarithm
Finding Common Logarithm
Solution

4. Definition of a Logarithm
Common Logarithm
Definition
A common logarithm is a logarithm with base We write log(a) to represent log10(a).
Rewrite each logarithmic expression as an exponential and simplify.
Example

5. Definition of a Logarithm
Common Logarithm
Solution

6. Properties of Logarithms
Properties and Definitions of Logarithms
Properties
For b > 0 and b ≠ 1,
logb (b) = 1
logb (1) = 0
A logarithmic function, base b, is a function that can be put into the form
where b> 0 and b ≠ 1.
Definitions

7. Properties of Logarithms
For an exponential function
For a logarithmic function
In words, are inverse functions.

8. Find the inverse of the function. 2.
Properties of Logarithms
Finding an Inverse Function
Example
Find the inverse of the function. 2.
Solution

9. Properties of Logarithms
Evaluating f and f-1
Example
Let
Solution

10. Graphing a Logarithmic Function
Example
Sketch the graph of
Applying the four-step method:
Step 1: Sketch a curve of f:
Step 2: Choose several points on the graph of f: (-1, 1/3), (0, 1), (1, 3) and (2, 9).
Solution

11. Graphing a Logarithmic Function
Solution
Example
Step 3: For each point (a, b) chosen in step 2, plot point (b, a): We plot (1/3, -1), (1, 0), (3, 1) and (9, 2).
Step 4: Sketch the curve containing the points in Step 3.

12. Using Logarithms to Model Authentic Situations
Logarithm Uses
Logarithm scales are used for:
Amplitudes or earthquakes
Noise levels of sounds
pH values
Earthquake Richter scale, R, is given by
where A is the amplitude and A0 is the reference amp.
Definition

13. Using Logarithms to Model Authentic Situations
Richter Numbers
Example
In 1906, an earthquake in San Francisco had an amplitude times the reference amplitude A0. In 1989, an earthquake had an amplitude times A0.
Find the Richter number of both earthquakes.
Find the ratio of the amplitudes of the 1906 and earthquakes.

14. 1. The Richter number of the 1906 earthquake is
Using Logarithms to Model Authentic Situations
Richter Numbers
Solution
1. The Richter number of the 1906 earthquake is
2. The Richter number of the 1989 earthquake is

15. Using Logarithms to Model Authentic Situations
Richter Numbers
Solution
Solution Continued
3. The ratio of the amplitudes is:
So, the 1906 earthquake had an amplitude 25
times greater than that of the 1989 earthquake.