Meaning of a logarithm | Year 11

01 / LEARN

The idea, step by step

A logarithm answers an exponent question

The statement ${log}_{b} a = c$ means $b^{c} = a$ . The base must be positive and not equal to 1. For real logarithms, the argument a must be positive.

The logarithm is not a multiplier or an abbreviation for division. For example, ${log}_{2} 8 = 3$ because the exponent needed on 2 to produce 8 is 3.

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

Base ten and inverse relationships

The calculator log key normally means base ten. In this chapter, an unsubscripted log denotes base ten. Thus $log 100 = 2$ and $log 0.01 = - 2$ . Values between zero and one have negative common logs.

Exponentiation and a matching logarithm undo one another on their valid domains: $b^{{log}_{b} x} = x$ for x>0. Check the base and argument before using an inverse relationship.

A common trap

Assuming a logarithm cannot be negative.

Instead: Its argument must be positive, but its value can be negative: a base-ten log of a number between 0 and 1 is negative.

02 / WORKED EXAMPLES

See the thinking

Example 1

Read an exact logarithm

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

Write the exponent question $3^{c} = 81$ .
Since $3^{4} = 81$ , the required exponent is 4.

Example 2

A logarithm of a small quantity

Solve $10^{x} = 0.004$ .

Take a base-ten logarithm: $x = log (0.004)$ .
Because $0.001 < 0.004 < 0.01$ , x must lie between -3 and -2.
A calculator gives $x \approx - 2.39794$ .

x \approx - 2.39794

Check your understanding

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

Show answer

$5^{3} = 125$

The logarithm records the exponent on the stated base.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF 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.
$10^{4} < N < 10^{5}$

Checking saved progress…

Continue learning

Logarithm laws

Next lesson

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 ↗

Back to Exponentials and logarithms