1. Logarithmic Functions
Section 4.2 Part 1
Logarithmic Functions
Objectives:
Change from logarithmic to exponential form.
Change from exponential to logarithmic form.
Evaluate logarithms.
Use basic logarithmic properties.

2. Definition of a Logarithmic Function
For x > 0 and b > 0, b = 1,
y = logb x is equivalent to by = x.
The function f (x) = logb x is the
logarithmic function with base b.

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

4. Practice #1 Write each equation in its equivalent exponential form.
a. 2 = log5 x b. 3 = logb 64 c. log3 7 = y

5. With the fact that y = logb x means by = x,
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.

6. Practice #2
Evaluate
a. log2 16 b. log c. log25 5
Solution

7. First, rewrite each expression as a logarithmic equation and convert
Solution
First, rewrite each expression as a logarithmic equation and convert
to an exponential equation.
Logarithmic Equation
Exponential Equation
Question Needed for Evaluation
Evaluation
a. log216 = x
2x = 16
2 to what power is 16? 4
b. log39 = x
3x = 9
3 to what power is 9? 2
c. log255 = x
25x = 5
25 to what power is 5?
Since you must take the square root of 25 to get 5, that is the same as an exponent of ½. The answer is ½ .

8. 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).

9. Inverse Properties of Logarithms
For x > 0 and b  1,
logb bx = x The logarithm with base b of b raised to a power equals that power.
b logb x = x b raised to the logarithm with base b of a number equals that number.

10. Practice #3
a. log1111 b. log446 c.