1. Logarithmic Functions and Their Graphs

2. This is a one-to-one function, therefore it has an inverse.
Consider
This is a one-to-one function, therefore it has an inverse.
The inverse is called a logarithm function.
Two raised to what power is 16?
Example:
The most commonly used bases for logs are 10:
and e:
is called the natural log function.
is called the common log function.

3. Definition of Logarithmic Function Logarithmic Form Exponential Form
b > 0; b  1
Logarithmic Form Exponential Form
y = logb x x = by

4. The log to the base “b” of “x” is the exponent to which “b” must be raised to obtain “x”
y = log10 x
 x = 10 y
y = log e x
 x = e y

5. Change from Logarithmic To Exponential Form
 8 = 23
Log = ½
 5 = 25 ½

6. Change from Exponential To Form Logarithmic
49 = 7 2
 log 7 49 = 2
1/5 = 5 –1
 log 5 (1/5) = -1

7. Properties of Logarithmic Functions
If b, M, and N are positive real numbers, b  1, and p and x are real numbers, then:
Log15 1 = 0
150 = 1
Log10 10 = 1
101 = 10
Log5 5x = x
5x = 5x
3log x = x 3

8. The Decibel Scale
The decibel level D of a sound of intensity I , measured in watts per square meter (W/ m2) is given by
where I0 = 10–12 W/ m2 is the intensity of the least audible sound that an average healthy person can hear.
Sound Intensity, W/ m2 Sound
  1.0  10–12 Threshold of hearing
5.2  10–10 Whisper
3.2  10–6 Normal conversation
8.5  10–4 Heavy traffic
3.2  10–3 Jackhammer
1.0  100 Threshold of pain
8.3  102 Jet plane with afterburner

9. The Richter Scale
The magnitude M on the Richter scale of an earthquake that releases energy E , measured in joules, is given by
where E0 = joules is the energy released by a small reference earthquake.
Magnitude on Richter scale Destructive power
M < 4.5 Small
4.5 < M < 5.5 Moderate
5.5 < M < 6.5 Large
6.5 < M < Major
7.5 < M Greatest

10. Since logs and exponentials are inverses the domain and range switch
Since logs and exponentials are inverses the domain and range switch!…the x values and y values are exchanged…

12. Logarithmic Function with Base 2y
= 2x y f x y
= 2 –3 1 8 –2 4 –1 2 3
Ordered
pairs
reversed y = x 10
f -1 x
= 2y 5 or y
= log x 2 x –5 5 10 –5
DOMAIN of f
= (– , ) = RANGE of
f -1 f
f -1
RANGE of
= (0, ) = DOMAIN of