# Complex Analysis/Elementary Functions/Logarithmic Functions

## Logarithmic Functions

edit## Logarithm

editA **logarithm** is the exponent that a base is raised to get a value. Such exponential equations can be written as logarithmic equations and vice versa. **Exponential equations** are in the form of b^{x} = a , and **logarithmic equations** are in the form of log_{b}a = x . When converting from exponential to logarithmic form, and vice versa, there are some key points to keep in mind:

1. The base of the exponent become the base of the logarithm.

Example:

3^{7} = 2187

log_{3}2187 = 7

2. The exponent is the logarithm.

Example:

5^{2} = 25

log_{5}25 = 2

3. Any nonzero base to the 0 power is 1.

6^{0} = 1

log_{6}1 = 0

4. An exponent or log can be negative.

4^{-2} = 0.0625

log_{4}0.0625 = -2

5. The exponent and the log can be variables.

4^{y} = 1024

log_{4}1024 = y

A logarithm is also an exponent. This means that the exponent rules apply to logarithms as well.

A **common logarithm** is a logarithm that has a base of 10. Bases of logarithms are known to be 10 when there is no base written for them. For example:

log6 = log_{10}6

Logarithmic functions are inverses of exponential functions, since logarithms are inverses of exponents. For example:

y = 3^{x}

is the inverse of

y = log_{3}x

And, since these two functions are inverses, their domain and ranges are switched. So, for

y = 3^{x}

the domain is all real numbers and the range is y > 0.

And, for

y = log_{3}x

the domain is x > 0 and the range is all real numbers.