Logarithms have a reputation for being difficult, and almost all of it belongs to the notation rather than to the idea. Strip the symbols away and a logarithm asks a single question: what exponent turns the base into this number?

Note The one agreement everything rests on

Every positive number can be written as a base raised to an exponent — provided we first agree on the base.

Fix the base at $2$ . Then:

8 = 2^{3}

3 = 2^{1.585\ldots}

\tfrac{1}{2} = 2^{-1}

1 = 2^{0}

The exponent is not always a whole number, and it does not need to be. Change the agreement to base $10$ and the same numbers simply get different exponents:

100 = 10^{2}

5 = 10^{0.699\ldots}

0.1 = 10^{-1}

1 = 10^{0}

Pick any positive number, fix any base, and the exponent exists. This is worth testing on a few numbers of your own, because everything that follows is built on it.

Definition The logarithm

A logarithm hands back that exponent. Writing $\log_{b} x$ is asking: to what power must $b$ be raised to give $x$ ?

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

The two statements carry exactly the same information, read from opposite ends. The exponential form starts from the exponent and produces the number; the logarithmic form starts from the number and produces the exponent.

The answer to a logarithm is always an exponent.

Note Why the domain stops at zero

The agreement was every positive number, and the word positive is doing real work.

There is no exponent that turns a positive base into $0$ . Raising $2$ to larger and larger negative powers gives $\tfrac{1}{2},\ \tfrac{1}{4},\ \tfrac{1}{8},\ \ldots$ — smaller and smaller, but never zero. Negative results are unreachable in the same way: a positive base raised to any real power stays positive.

Since no exponent produces them, zero and the negative numbers are simply left out of the domain. $\log_{b} x$ is defined for $x > 0$ .

Note Reading the exponent off the graph

Graphing $y = x$ is easy: move along $x$ , and $y$ matches it. Graphing $y = 2^{x}$ is nearly as easy: move along $x$ , and $y$ is $2$ raised to that power, so $1 \mapsto 2$ , $2 \mapsto 4$ , $3 \mapsto 8$ .

The logarithm runs the same machine backwards. Move along $x$ , and ask what exponent turns the base into it. That exponent is $y$ .

x = 1

the exponent that gives

1

0

, so the curve crosses the

x

-axis there. At

x = 2

the exponent is

1

. At

x = 4

it is

2

The dashed line $y = x$ makes the relationship visible: the logarithm is the exponential curve reflected across it. Same pairs of numbers, swapped ends.

Example Reading three logarithms

In each case, name the base first, then ask for the exponent.

\log_{2} 8

— what power of

2

gives

8

? Since

2^{3} = 8

⟹ $\log_{2} 8 = 3$

\log_{10} 0.1

— what power of

10

gives

0.1

? Since

10^{-1} = 0.1

⟹ $\log_{10} 0.1 = -1$

\log_{5} 1

— what power of

5

gives

1

? Since

5^{0} = 1

⟹ $\log_{5} 1 = 0$

The last one holds for every base, because any base raised to $0$ gives $1$ . That is why every logarithmic curve passes through the point $(1,\, 0)$ .

Note What changing the base does

Before anything else, a logarithm needs a base: $2$ , $10$ , the number $e$ , or any other. Change it and the shape survives, but the climb changes.

All three cross at

(1,\, 0)

, because

b^{0} = 1

whatever

b

is.

Base $2$ climbs fastest — it only takes $3$ doublings to reach $8$ . Base $10$ climbs slowest, since a single step of $1$ in $y$ demands ten times the $x$ . Base $e$ sits between them, and the reason that particular number matters so much is a lesson of its own.

Summary

Every positive number can be written as a base raised to an exponent, once a base is agreed on.
$\log_{b} x$ is the exponent that turns $b$ into $x$ .
$\log_{b} x = y$ and $b^{y} = x$ say the same thing from opposite ends.
Ask for the base before anything else — without it the question is incomplete.
Zero and negative numbers are outside the domain, because no exponent reaches them.
$\log_{b} 1 = 0$ in every base, so every curve passes through $(1,\, 0)$ .
A larger base makes the curve climb more slowly: $2$ is fastest, $e$ is between, $10$ is slowest.

Introduction to Logarithms