What is a logarithm / What are logarithms

Logarithms

An old logarithm table and modern calculator
An old logarithm table and modern calculator

Logarithms come in the form \({\log _a}x\). We say this as 'log to the base \(a\) of \(x\). But what does \({\log _a}x\) mean?

Example

1) \({\log _5}25\) means "What power of \(5\) gives \(25\)?""

The answer is \(2\) because \({5^2} = 25\), in other words \({\log _5}25 = 2\).

Question

2) \({\log _2}16\) means, What power of \(2\) gives \(16\)?

The answer is \(4\) because \({2^4} = 16\), in other words \({\log _2}16 = 4\).

So \({\log _a}x\) means "What power of \(a\) gives \(x\)?" Note that both \(a\) and \(x\) must be positive.

You may wish to use these to help remember this:

curriculum-key-fact
  • \[{\log _a}x=y\]
  • \[x=a^{y}\]