**Logarithms** are algebraic concepts that complete the "exponential circle" (depicted to right), a metaphor for the three variables in a generic exponential expression.

With the use of logarithms, it is possible to solve for any one variable in terms of the other two.

Note how the blue text and arrows indicate exponentiation, while the red indicates the typical reverse of that — roots. The logarithmic relationship, in green, illustrates how one may be resolved in terms of the others.

This is what logarithms do, algebraically. It is a highly useful tool in rearranging and manipulating algebraic expressions.

The subscripted number is the base of the logarithm, because it is also the base of the exponential.

## Example

Knowing that $ 3^2=9 $ , and therefore that $ \sqrt[2]{9}=3 $ , allows us to express the same three numbers with a single exponential relationship in two different ways, one solved for 9 and the other solved for 3.

But if you wanted to solve for the 2, or any such exponent, you must do so with logarithms: $ \log_3(9)=2 $

Likewise, as the image above illustrates, if $ a^b=c $ then $ \sqrt[b]{c}=a $ and $ \log_a(c)=b $ .

It is now possible to solve equations in the form $ 3^{2x}=81 $ without having to guess and check, there is now a systematic and algebraic method of arriving at the proper solution explicitly.

## Restrictions

Logarithms with the base of 0 or 1 are undefined.

Logarithms of negative values are non-real complex numbers. (see complex analysis).

## A note on logarithms

Logarithms require a base number, because, after all, the function has two inputs which are related to one another in a specific way, producing a single output. It would be impossible to arrive at a single solution knowing only the input or the base but not both. Therefore, all logarithms require both the inputted value and its base subscript. Logarithms are a dual-input, single-output function.

However, it is a commonly accepted convention to neglect writing a subscripted base when referring to base ten logarithms. Logarithms of base ten can be written in one of three ways, and is specially named the common logarithm.

- $ \log(a)=\log_{10}(a) $
- $ \lg(a)=\log_{10}(a) $ (Common to Engineers)

Furthermore, another exception to the rule is the natural logarithm, which uses Euler's number (denoted $ e $) as the base. It, too, can be written in two ways:

- $ \ln(a)=\log_{e}(a) $

## Graphing logarithmic functions

All logarithms pass through the point $ (1,0) $ and have an asymptote of $ x=0 $ . In addition,

- $ \lim_{x\to\infty}\log_a(x)=\infty $

## Inverse exponentiation

The logarithm is also the inverse function of an exponential function having the same base.

Given the two functions:

- $ y=f(x)=b^x $
- $ y=f^{-1}(x)=\log_b(x) $

Each is the inverse of the other. On a graph they will be perfect reflections of one another across the line $ y=x $ .

Exponentiation, being the reverse operation of finding the logarithm, is sometimes referred to as the antilog operation.

## Properties of logarithms

- $ \log_a(xy)=\log_a(x)+\log_a(y) $
- $ \log_a\left(\frac{x}{y}\right)=\log_a(x)-\log_a(y) $
- $ \log_a(x^n)=n\log_a(x) $
- $ \log_a(x)=\frac{\log_b(x)}{\log_b(a)} $