1. 5.4 Logarithmic Functions and Models
Evaluate the common logarithmic function
Evaluate logarithms with other bases
Solve basic exponential and logarithmic equations
Solve general exponential and logarithmic equations
Convert between exponential and logarithmic forms

2. Common Logarithm (1 of 3)

3. Common Logarithm (2 of 3)

4. Common Logarithm (3 of 3)
The common logarithm of a positive number x, denoted log x, is defined by
where k is a real number. The function given by f(x) = log x is called the common logarithmic function.

5. Evaluate

6. Example: Evaluating common logarithms
Simplify each logarithm by hand.

7. Graphs of f(x) = 10x and f –1(x) = log x
Their graphs are reflections across the line y = x.

8. Inverse Properties of the Common Logarithm
The following inverse properties hold for the common logarithm.

9. Base-2 Logarithm x 2−3.1 2−2 2−0.5 20 20.5 22 23.1 log2x −3.1 −2 −0.5
0.5 2
3.1
A base-2 logarithm is an exponent of 2.

10. Base-e Logarithm - Natural Logarithmx
e−3.1
e−2
e−0.5 e0
e0.5 e2
e3.1
ln x
−3.1 −2
−0.5
0.5 2
3.1
A natural logarithm is an exponent of e.

11. Logarithm

12. Example: Evaluating logarithms
Evaluate each logarithm.

13. Inverse Properties
The following inverse properties hold for logarithms with base a.

14. Example: Applying inverse properties (1 of 2)
Use inverse properties to evaluate each expression.

15. Example: Applying inverse properties (2 of 2)

16. Example: Solving a base-10 exponential equation

17. Example: Solving a common logarithmic equation

18. Example: Solving exponential equations

19. Example: Solving logarithmic equations