Math Notes Logarithmic Functions

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  • Topic: Natural logarithm, Logarithm, E
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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

Solution:

4 4
3

b 

b  64

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

logb 81  2
log b 81  2 b  81
2

Solution:

b 
2

1/2

 81

1/2

b  1/ 9

Logarithmic Function with base b
Recall:
f  f 1  x    x f 1  f  x    x

Since the logarithmic function with base b is the inverse of exponential function with base b, then

log b b  x
x

b

logb x

x

Logarithmic Function with base b
Example:

log 2 2  5 log10 10  3
3

5

Graph of Logarithmic function of base b
Since logarithmic function is the inverse of exponential function, then its graph is the reflection of the latter function at y=x.

b 1

y  logb x

y b

x

0  b 1

y b

x

y  logb x

Natural Logarithmic Function
Definition: The natural logarithmic function is the inverse of the natural exponential function.

y  ln x

if and only if


xe

y

Dom  f  

Rng  f  

Natural Logarithmic Function
Note:

ln e  1

e

ln x

x
x

ln e  x

Properties of Logarithmic Functions
Theorem: If b>0, b≠1, and u and v are positive numbers, then

logb uv  logb u  logb v
u log b  log b u  log b v v

Properties of Logarithmic Functions
Theorem: If b>0, b≠1, and u and v are positive numbers, then

logb u  n logb u
n

Properties of Logarithmic Functions
Examples: Express each of the
following in terms of log of x, y and z, each of which represents a positive number.

log b x y z

2

3 4

x log b 2 yz

log b

5

xy 3 z

2

Properties of Logarithmic Functions
Examples: Write each of the
following expression as a single logarithm with a coefficient of 1.

logb x  2logb y  3logb z
1  logb 4  logb 3  logb x  logb y  3

TOPICS for TODAY
• Exponential and Logarithmic Equations • Application • EXERCISE

RULES FOR SOLVING LOGARITHMIC AND EXPONENTIAL EQUATIONS • Use log of base 10...
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