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
x9
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
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
x9
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