Meaning of a logarithm | Year 11

01 / LEARN

Understand the idea.

A logarithm answers an exponent question

A logarithm answers an exponent question: ${log}_{b} a$ asks which power of $b$ produces $a$ . Rewriting in exponential form often makes an exact value immediate. The logarithm is a number, rather than a multiplication instruction.

For real logarithms, require $b > 0$ , $b \neq 1$ and $a > 0$ . A positive base different from 1 gives a one-to-one exponential function, so each permitted argument corresponds to exactly one exponent.

{log}_{b} a = c b^{c} = a

Base ten and inverse relationships

Here an unsubscripted logarithm means base ten, matching the usual calculator log key. Powers of ten provide reference values: $log 100 = 2$ and $log 0.01 = - 2$ . A positive argument between zero and one has a negative base-ten logarithm.

A logarithm and exponentiation undo one another when their bases match. Thus $b^{{log}_{b} x} = x$ for $x > 0$ , while ${log}_{b} (b^{u}) = u$ for every real $u$ . Check the base and domain before using either identity.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Read an exact logarithm

Find ${log}_{3} 81$ .

Worked solution

1
Name the required exponent.
${log}_{3} 81 = c ⟺ 3^{c} = 81$
The logarithm records the exponent on base 3.
2
Express the argument as a power of 3.
$81 = 3^{4}$
Four factors of 3 multiply to give 81.
3
Match the exponents.
$c = 4 ⟹ {log}_{3} 81 = 4$
The exponential with base 3 is one-to-one.

Answer

EXAMPLE 02

A logarithm of a small quantity

Solve $10^{x} = 0.004$ .

Worked solution

1
Use the matching base-ten inverse.
$10^{x} = 0.004 ⟹ x = log (0.004)$
The argument is positive, so its real common logarithm exists.
2
Locate the exponent before calculating.
$10^{- 3} = 0.001 < 0.004 < 0.01 = 10^{- 2} ⟹ - 3 < x < - 2$
A base-ten exponential increases with its exponent.
3
Evaluate and check the sign.
$x = log (0.004) \approx - 2.39794$
The result lies between the reference exponents, consistent with an argument below 1.

Answer

$x \approx - 2.39794$

03 / YOUR TURN

Try it yourself.

Check your understanding

Write ${log}_{5} 125 = 3$ in exponential form.

Show answer & reasoning

$5^{3} = 125$

The base is 5 and the recorded exponent is 3. Exponential form is $5^{3} = 125$ ; indeed, multiplying three factors of 5 gives 125.

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Evaluate ${log}_{2} 32$ .
2 marks
Evaluate $log 0.001$ .
2 marks
Write $7^{2} = 49$ as a logarithmic statement.
2 marks
Evaluate $10^{log 7}$ .
2 marks
Explain why $log 0$ is not a real number.
3 marks
If $log N = 4.2$ , between which consecutive powers of ten does $N$ lie?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Evaluate ${log}_{2} 32$ .
2 marks
$2^{5} = 32$ .
5
Evaluate $log 0.001$ .
2 marks
$10^{- 3} = 0.001$ .
-3
Write $7^{2} = 49$ as a logarithmic statement.
2 marks
The exponent 2 becomes the value of the logarithm.
${log}_{7} 49 = 2$
Evaluate $10^{log 7}$ .
2 marks
The exponential and its base-ten logarithm are inverse operations for positive inputs.
7
Explain why $log 0$ is not a real number.
3 marks
$10^{x} > 0$ for every real $x$ ; it approaches zero without reaching it.
No real exponent on 10 gives zero.
If $log N = 4.2$ , between which consecutive powers of ten does $N$ lie?
3 marks
$4 < 4.2 < 5$ and the function $10^{x}$ increases.
$10^{4} < N < 10^{5}$

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: exponential and logarithmic functions, growth models and numerical/algebraic prerequisites.

Official curriculum source ↗

In this lesson

Understand the idea.

A logarithm answers an exponent question

Base ten and inverse relationships

See the method in action.

Read an exact logarithm

Name the required exponent.

Express the argument as a power of 3.

Match the exponents.

A logarithm of a small quantity

Use the matching base-ten inverse.

Locate the exponent before calculating.

Evaluate and check the sign.

Try it yourself.

Check your understanding

Keep practising