YEAR 10 · EXPONENTIAL RELATIONSHIPS · 15 MIN

Logarithms as inverses

Understand a logarithm as the exponent needed to produce a positive number.

In this lesson

Convert log statements into exponential statements
Respect the domain of a real logarithm

Before you start

Exponential notation
Inverse operations

01 / LEARN

Understand the idea.

A logarithm answers an exponent question

A logarithm asks for an exponent: ${log}_{b} a = c$ means exactly $b^{c} = a$ . The base must satisfy $b > 0$ and $b \neq 1$ , and the real logarithm input must satisfy $a > 0$ .

Read ${log}_{2} 8$ as the power on $2$ that produces $8$ . A logarithm can return zero, positive or negative values: for example, ${log}_{2} 1 = 0$ and ${log}_{2} (\frac{1}{2}) = - 1$ .

Undoing exponential functions

Exponentials and logarithms of the same base are inverses: ${log}_{b} (b^{x}) = x$ for real $x$ , and $b^{{log}_{b} a} = a$ for $a > 0$ . Keep the base unchanged when moving between the two forms.

On a scientific calculator, the key labelled log usually means base $10$ , while ln means base $e$ . State the intended base rather than assuming different logarithms give the same output.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Convert notation

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

Worked solution

1
Translate the logarithm into an exponent question.
${log}_{3} 81 = c ⟺ 3^{c} = 81$
The logarithm is the exponent on the stated base.
2
Express the input as a power of three.
$81 = 3^{4}$
Four factors of $3$ multiply to $81$ .
3
Read the required exponent.
${log}_{3} 81 = 4$
The unique exponent producing $81$ is $4$ .

Answer

EXAMPLE 02

A negative logarithm

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

Worked solution

1
Express the decimal as a power of ten.
$0.001 = \frac{1}{1000} = 10^{- 3}$
Three places below the units place corresponds to the exponent $- 3$ .
2
Apply the inverse relationship.
${log}_{10} (10^{- 3}) = - 3$
A base- $10$ logarithm recovers the exponent of a power of $10$ .
3
Check the sign using the input.
$10^{- 3} = 0.001 > 0$
The negative answer is allowed; it is the logarithm input, rather than its output, that must be positive.

Answer

−3

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$5^{3} = 125$

The statement says the exponent on base $5$ is $3$ . Therefore its exponential form is $5^{3} = 125$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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 $4^{- 2} = \frac{1}{16}$ 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 $4^{- 2} = \frac{1}{16}$ in logarithmic form.
2 marks
The log output is the exponent $- 2$ .
${log}_{4} (\frac{1}{16}) = - 2$
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 ↗

In this lesson

Understand the idea.

A logarithm answers an exponent question

Undoing exponential functions

See the method in action.

Convert notation

Translate the logarithm into an exponent question.

Express the input as a power of three.

Read the required exponent.

A negative logarithm

Express the decimal as a power of ten.

Apply the inverse relationship.

Check the sign using the input.

Try it yourself.

Check your understanding

Keep practising