The Meaning of a Logarithm

A logarithm is not a new operation to memorise — it is a question. Every time you meet one, ask the same thing: what power must the base be raised to in order to give this number? The answer to that question is the logarithm. Everything else in the topic follows from it.

Concept The three parts, and the general form

\log_b(x) = y \quad \Longleftrightarrow \quad b^y = x

These two statements say exactly the same thing in different notation. A logarithm relates three quantities:

b

— the base, the number being raised to a power.

x

— the result, the number you are trying to reach.

y

— the exponent, which is what the logarithm actually gives you.

Reading

\log_b(x)

aloud as "how many times do I raise

b

to reach

x

?" turns every logarithm problem into an exponent problem you already know how to solve.

Example Evaluating log₂(8)

Find $\log_2(8)$ .

Ask the question: 2 raised to what power gives 8?

Try the powers of 2:

2^1 = 2

2^2 = 4

2^3 = 8

The exponent that works is 3.

⟹ log₂(8) = 3

Example Evaluating log₃(27)

Find $\log_3(27)$ .

Ask: 3 raised to what power gives 27?

3^1 = 3

3^2 = 9

3^3 = 27

⟹ log₃(27) = 3

Both answers came out as 3, but for different reasons — the bases and results were different. The logarithm is never just the result; it is the exponent attached to that base.

Concept Why a base of 1 is not allowed

Consider $\log_1(5)$ . Asking the usual question gives: 1 raised to what power gives 5? Work through the powers of 1:

1^1 = 1

1^2 = 1

1^{10} = 1

1^{100} = 1

No matter which exponent you choose, the answer is always 1 — you can never reach 5. There is no exponent that works, so $\log_1(5)$ is undefined. This is the same reason a base of 1 was excluded from exponential functions.

Concept The conditions on the base

b > 0 \quad \text{and} \quad b \neq 1

The base must be positive, and it must not equal 1. A base of 1 produces only the value 1, as shown above. A negative base would jump between positive and negative values as the exponent changed, so it could not reliably reach a given result either. These two conditions are what keep a logarithm well defined.

Example A few quick evaluations

\log_5(25) = 2

, because

5^2 = 25

\log_{10}(1000) = 3

, because

10^3 = 1000

\log_2(1) = 0

, because

2^0 = 1

— and this holds for any valid base.

\log_7(7) = 1

, because

7^1 = 7

\log_2\!\left(\frac{1}{2}\right) = -1

, because

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

The last two are worth remembering as patterns: the log of the base itself is always 1, and the log of 1 is always 0.

Note Mistakes to avoid

Reading

\log_2(8)

as 8 rather than as the exponent 3 — the logarithm is always the exponent.

Mixing up the base and the result: in

\log_b(x)

, the small subscript is the base.

Using a base of 1, or a negative base, both of which leave the logarithm undefined.

Expecting

\log_b(0)

to have a value; no power of a positive base ever gives 0.

Forgetting that

\log_b(1) = 0

for every valid base, since

b^0 = 1

Summary

A logarithm answers one question: what power must the base be raised to in order to give this number?
log_b(x) = y and bʸ = x are two ways of writing the same statement.
The logarithm is the exponent — so log₂(8) = 3 and log₃(27) = 3.
A base of 1 is undefined, because every power of 1 is still 1.
For a logarithm to be defined the base must satisfy b > 0 and b ≠ 1.