Defining the Logarithm

Idea

$2^3 = 8$ answers "what do I get?". Turn it around and ask "what power?" instead: $2$ to what power gives $8$ ?

A logarithm is the answer to that question, and nothing more.

Rule

$\log_a x = y \iff a^y = x$

$a > 0, \quad a \ne 1, \quad x > 0$

Read the notation out loud. $\log_a x$ is the power that turns $a$ into $x$ .

↳ $\log_2 8 = 3$ because $2^3 = 8$

↳ $\log_5 25 = 2$ because $5^2 = 25$

↳ $\log_{10} 1000 = 3$ because $10^3 = 1000$

How it is used

1. Every logarithm is an index statement in disguise

The two forms carry the same information. Being able to swap them on sight is most of the skill.

Index form	Logarithm form
$a^y = x$	$\log_a x = y$
$3^4 = 81$	$\log_3 81 = 4$
$2^{-3} = \dfrac{1}{8}$	$\log_2 \dfrac{1}{8} = -3$
$9^{1/2} = 3$	$\log_9 3 = \dfrac{1}{2}$

The base stays the base in both. Only the other two numbers change places.

2. Two values you never have to work out

\log_a 1 = 0

\log_a a = 1

Because $a^0 = 1$ and $a^1 = a$ , for every allowed base. They hold whatever $a$ is.

3. The argument must be positive

$a^y$ is always positive, so $x$ is always positive. There is no power of $2$ that gives $-8$ , or $0$ .

$\log_2(-8)$ has no value

$\log_2 0$ has no value

That is why the domain of $\log_a x$ is $x > 0$ , and it is the first thing to check in any question about a domain.

4. It is the exponential, read backwards

The logarithm undoes the exponential, and the exponential undoes the logarithm.

\log_a(a^x) = x

a^{\log_a x} = x

On a graph the two curves are mirror images in the line $y = x$ . Nothing has changed but which axis you read from.

Exponential and logarithm

Try this. Compare bases 2 and 0.5: growth becomes decay. Move the input to 1; the logarithm is zero for either base.

BasePositive input x0.5

Base 2 · Blue: y = 2ˣ · Green: y = log_b(x) · Dashed: y = x. At x = 0.5, y = -1.

Worked example

Evaluate $\log_2 32$ .

Remember

$\log_a x$ asks: $a$ to what power gives $x$ $2^1 = 2$ , $2^2 = 4$ , $2^3 = 8$ , $2^4 = 16$ , $2^5 = 32$

Read it as a question. Which power of $2$ gives $32$ ?

2^{\;?} = 32

Count up the powers of $2$ until you reach it.

2, \; 4, \; 8, \; 16, \; 32

That is the fifth one.

\log_2 32 = 5

Check by going back to index form: $2^5 = 32$ .

Explore

Drag the probe and watch the two marked points. One sits on $y = a^x$ , the other on $y = \log_a x$ , and their coordinates are the same pair in the opposite order.

That swap is the definition. Change the base with the slider and the pair still swaps, whatever base you choose.