Math Notes Logarithmic Functions

Exponential and Logarithmic Functions
2.2 Logarithmic Functions MATH14

• Logarithmic Function with base b • Graph of Logarithmic Function • Natural Logarithmic Function • Properties of Logarithmic Functions • Exponential and Logarithmic Equations

Logarithmic Function with base b
Definition: The logarithmic function with base b is the inverse of the exponential function with base b.

y  logb x
Note: Dom  f  

if and only if


x b

y

Rng  f  

Logarithmic Function with base b
Examples:

3 9 1/2 1 1    4  16 
2




log 3 9  2 1 1 log1/16  4 2

2 8
3

1 5   25
2

Logarithmic Function with base b
More examples: Find the values of the ff:
Solution:

log 7 49 log 7 49  y 7  49 y y

Therefore, log 7 49  2

7 7

2

Logarithmic Function with base b
More examples: Find the values of the ff:
Solution: log 5  y 5 y y

log 5 5
5  5 5 5
1/2

Therefore,log 5

5  1/ 2

Logarithmic Function with base b
More examples: Find the values of the ff:

1 log 6 6 Solution: log 1/ 6  y 6 y y

6  1/ 6 6 6
1

Therefore,log 6 1/ 6  1

Logarithmic Function with base b
More examples: Find the values of the ff:
Solution:

log3 81 log 3 81  y

3  81 y y

3 3
Therefore,

4

log3 81  4

Logarithmic Function with base b
More examples: Find the values of the ff:

log10 0.001
Solution:
log10 0.001  y 10 y  0.001 10  10 y 3

Therefore,log10 0.001  3

Logarithmic Function with base b
Other examples:
Solve the given equation for either x or b.

log 6 x  2
6 x
2

Solution:

log 6 x  2 x  36

Logarithmic Function with base b
Other examples:
Solve the given equation for either x or b.

log 27 x  2 / 3
27 2/3  x x

Solution:

log 27 x  2 / 3



3

27



2

x9

Logarithmic Function with base b
Other examples:
Solve the given equation for either x or b.

logb 4  1/ 3 log b 4  1/ 3 b
1/3 1/3 3