Logarithms as inverses | Year 10

01 / LEARN

The idea, step by step

A logarithm answers an exponent question

${log}_{b} a = c$ means $b^{c} = a$ . The base must satisfy $b > 0$ , $b \neq 1$ , and the logarithm input must be $a > 0$ .

For example, ${log}_{2} 8 = 3$ because $2^{3} = 8$ . A logarithm may be negative even though its input must be positive.

Undoing exponential functions

Exponentiation and logarithms of the same base undo each other: ${log}_{b} (b^{x}) = x$ and $b^{{log}_{b} a} = a$ for valid inputs.

When a calculator shows log without a base, it commonly means base $10$ . Always state the base if it affects the question.

A common trap

A logarithm cannot have a negative answer.

Instead: The input must be positive, but the output can be negative. Inputs between $0$ and $1$ have negative logs when the base is greater than $1$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Convert notation

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

The question asks which power of $3$ gives $81$ .
Since $3^{4} = 81$ , the logarithm is $4$ .

Example 2

A negative logarithm

Evaluate ${log}_{10} (0.001)$ .

$0.001 = 10^{- 3}$ .
Therefore the exponent is $- 3$ .

−3

Check your understanding

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

Show answer

$5^{3} = 125$

The logarithm gives the exponent on base $5$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Evaluate ${log}_{2} 16$ .
2 marks
Evaluate ${log}_{7} 1$ .
2 marks
Evaluate ${log}_{10} 100000$ .
2 marks
Write $4^{- 2} = 1 / 16$ in logarithmic form.
2 marks
Give the real domain of ${log}_{3} (x - 4)$ .
3 marks
Explain why ${log}_{2} 0$ is undefined as a real number.
3 marks

Read the worked solutions

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

Evaluate ${log}_{2} 16$ .
2 marks
$2^{4} = 16$ .
4
Evaluate ${log}_{7} 1$ .
2 marks
$7^{0} = 1$ .
0
Evaluate ${log}_{10} 100000$ .
2 marks
$10^{5} = 100000$ .
5
Write $4^{- 2} = 1 / 16$ in logarithmic form.
2 marks
The log output is the exponent $- 2$ .
${log}_{4} (1 / 16) = - 2$
Give the real domain of ${log}_{3} (x - 4)$ .
3 marks
The input must be positive: $x - 4 > 0$ .
$x > 4$
Explain why ${log}_{2} 0$ is undefined as a real number.
3 marks
$2^{x}$ approaches but never reaches zero, so no real exponent gives the required input.
No real power of 2 is zero.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Integer and fractional index laws; connecting fractional powers with roots; simplifying surds; operations with surds; rationalising denominators; exact and approximate values; exponential tables and graphs; growth/decay factors; comparing linear/quadratic/exponential patterns; exponential equations; logarithms as inverses; logarithm laws; interpreting logarithmic scales; graph transformations and parameter experiments; choosing and evaluating models.

Official curriculum source ↗

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