Complex Analysis/Elementary Functions/Logarithmic Functions

Logarithm
A 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 bx = a, and logarithmic equations are in the form of logba = 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:

37 = 2187

log32187 = 7

2. The exponent is the logarithm.

Example:

52 = 25

log525 = 2

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

60 = 1

log61 = 0

4. An exponent or log can be negative.

4-2 = 0.0625

log40.0625 = -2

5. The exponent and the log can be variables.

4y = 1024

log41024 = 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 = log106

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

y = 3x

is the inverse of

y = log3x

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

y = 3x

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

And, for

y = log3x

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