Logarithms : Maths Notes from Mr Shelswell

Logarithms are a way of expressing powers.

The logarithm $\log_b{x}$ for a base $b$ and a number $x$ is defined to be the inverse function of taking $b$ to the power $x$ , i.e., $b^x$ .¹

The simplest way to explain the concept of logarithms is through an example:

$\begin{aligned}2^5&=32 \\ \log_2{32}&=5\end {aligned}$

The subscript of the logarithm above denotes the base of the logarithm. This is equivalent to the number which is raised to a power in the preceding equation. The logarithm to the base 2 of 32 is 5, because 2 to the power of 5 is 32.

The equation $\log_3{81}=x$ can be solved by asking ourselves “what power of 3 gives a result of 81?” In this case the answer will be $x=4$ and can be calculated relatively easily. For most questions involving logarithms you will be resorting to the functions in your calculator or spreadsheet (or log tables, which have become increasingly rare).

$x^y = z \iff \log_x{z} = y$

¹ Weisstein, Eric W. “Logarithm.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Logarithm.html