1. Logarithmic Functions
Logarithms are numbers used to measure the severity of earthquakes, the loudness of sounds, and the brightness of stars.
The common logarithm of an integral power of 10 is its exponent

2. In general, the common logarithm of any positive real number x is defined to be the exponent you get when you write x as a power of 10

3. Every sound has an intensity level due to the power of the sound wave.
The unit for measuring the loudness of a sound is the decibel
The decibel is related to the intensity of sound by
A loud stereo set has an intensity level of
and is perceived to have a decibel level of

4. Many people think that when the intensity of a sound is doubled, the decibel level is also doubled. The following example shows that this is not so.

5. Example 1. Two loud stereos are playing the same music simultaneously at 80 dB each. What is the decibel level of the combined sound? By how many decibels is the decibel level of the two stereos greater than the decibel level of one stereo?

6. The decibel scale is an example of a logarithmic scale
The decibel scale is an example of a logarithmic scale. Such a scale is also used to measure acidity and brightness.
Common logs have base 10
Logarithms to other bases are sometimes used.
if and only if
Example. 2. Find

7. The base b logarithmic function is the inverse of the base b exponential function.
Domain of
All reals
Range of
Positive reals
Domain of
Positive reals
Range of
All reals
The most important logarithmic function in advanced mathematics and statistics has the number e as its base.
The natural logarithm of x is usually denoted ln x although sometimes it is written
if and only if

8. Example 3. Find the value of x to the nearest hundredth.