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 b^x = a , and logarithmic equations are in the form of log_ba = 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.

4^y = 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 = 3^x

is the inverse of

y = log₃x

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 = log₃x

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