What Is a Logarithm? Logs Explained Simply

Q: What does log base 2 of 8 mean?

It asks what power of 2 gives 8. Since 2^3 = 8, log_2 8 = 3.

What a logarithm actually asks, how to switch between log form and exponent form, the values every log shares, why you cannot take the log of a negative number, and how the log graph mirrors the exponential.

3 min read Supports 4 lessons

You know how to answer $2^3 = {?}$ . A logarithm answers the opposite question: $2^{?} = 8$ . The missing exponent is $3$ , so we write $\log_2 8 = 3$ , read "log base 2 of 8 is 3".

Key idea

A logarithm is an exponent. Whenever you see $\log_b x$ , ask: $b$ to what power gives $x$?

Definition

\log_b x = y \quad\Longleftrightarrow\quad b^{y} = x

Switching forms

Every log statement can be rewritten as an exponent statement and back again. The base stays the base.

$\log_3 81 = \;?$
3 to what power gives 81?

1/4

\log_{10} 1000 = 3 \qquad \log_5 \tfrac{1}{25} = -2 \qquad \log_9 3 = \tfrac12

Logs can be negative or fractional, because exponents can.

Values every log shares

$\log_b 1 = 0$ , because $b^0 = 1$ . And $\log_b b = 1$ , because $b^1 = b$ . These hold for every base.

Two bases are so common they get their own shorthand. $\log x$ usually means $\log_{10} x$ , and $\ln x$ , the natural log, means $\log_e x$ .

Logs undo exponentials

Taking $\log_b$ undoes raising to the power $b$ , and the other way round:

\log_b\!\left(b^x\right) = x \qquad\qquad b^{\log_b x} = x

So the log function is the inverse of the exponential function, and its graph is the exponential's reflection in the line $y = x$ .

Inverse functions: reflections in $y = x$ .

Watch out

You cannot take the log of zero or a negative number. No power of a positive base gives $0$ or anything negative, so $\log_2(-4)$ has no real value.

Why logs are useful

Logs turn huge ranges into manageable ones. Earthquake magnitudes, sound levels in decibels and the pH of a solution are all logarithmic scales: one step up means ten times more. And whenever the unknown is an exponent, as in $2^x = 20$ , a logarithm is the tool that gets it down.

Worked example

Solve $2^x = 20$ .

1
Rewrite in log form.
$x = \log_2 20$

Common questions

What does log base 2 of 8 mean?: It asks what power of $2$ gives $8$ . Since $2^3 = 8$ , $\log_2 8 = 3$ .
Why can't you take the log of a negative number?: A positive base raised to any real power is always positive. No exponent can produce a negative number or zero, so those logs are undefined.

Practise logarithms

The Logarithms lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

What does log base 2 of 8 mean?

It asks what power of $2$ gives $8$ . Since $2^3 = 8$ , $\log_2 8 = 3$ .

Why can't you take the log of a negative number?

A positive base raised to any real power is always positive. No exponent can produce a negative number or zero, so those logs are undefined.