Interpreting logarithmic scales | Year 10

01 / LEARN

The idea, step by step

Equal spacing need not mean equal addition

On a base-10 logarithmic axis, the labels $1, 10, 100, 1000$ have equal spacing. Moving one step multiplies by $10$ rather than adding a fixed number.

This makes quantities across many orders of magnitude readable. A graph must identify whether its axis is linear or logarithmic.

Use the stated definition

For a simplified amplitude scale $M = {log}_{10} (A / A_{0})$ , a difference of $d$ units corresponds to an amplitude ratio $10^{d}$ . This statement concerns amplitude, not earthquake energy.

Another example is a classroom concentration model $p = - {log}_{10} c$ . Increasing $p$ by $1$ divides the positive concentration $c$ by $10$ . The minus sign reverses the direction.

A common trap

A difference of 2 on a base-10 logarithmic scale means twice as large.

Instead: It means a factor of $10^{2} = 100$ . Compare the underlying quantities, not just the scale labels.

02 / WORKED EXAMPLES

See the thinking

Example 1

Read a ratio

On a base-10 log scale, one point is at $10^{2}$ and another at $10^{5}$ . How many times larger is the second quantity?

Divide: $10^{5} / 10^{2} = 10^{3}$ .
The ratio is $1000$ , even though the exponent labels differ by only $3$ .

1000 times

Example 2

Use a supplied formula

If $p = - {log}_{10} c$ and $c = 10^{- 4}$ , find $p$ .

${log}_{10} (10^{- 4}) = - 4$ .
The negative sign gives $p = 4$ .

Check your understanding

What ratio corresponds to a difference of 1.5 on a base-10 log scale?

Show answer

$10^{1.5} \approx 31.6$

Exponentiate the scale difference.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

What quantity sits halfway between $10$ and $1000$ on a base-10 logarithmic axis?
2 marks
How many equal log intervals separate $1$ and $10000$ when adjacent labels differ by a factor of 10?
2 marks
For $M = {log}_{10} (A / A_{0})$ , find $M$ when $A = 1000 A_{0}$ .
2 marks
Two amplitudes have scale values 4 and 6 under $M = {log}_{10} (A / A_{0})$ . Find their larger-to-smaller ratio.
2 marks
For $p = - {log}_{10} c$ , find $c$ when $p = 5$ .
2 marks
Why can zero not appear as a value on an ordinary real logarithmic axis?
3 marks

Read the worked solutions

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

What quantity sits halfway between $10$ and $1000$ on a base-10 logarithmic axis?
2 marks
The midpoint exponent is $(1 + 3) / 2 = 2$ , so the value is $10^{2}$ .
100
How many equal log intervals separate $1$ and $10000$ when adjacent labels differ by a factor of 10?
2 marks
$10000 = 10^{4}$ .
4
For $M = {log}_{10} (A / A_{0})$ , find $M$ when $A = 1000 A_{0}$ .
2 marks
$M = {log}_{10} 1000 = 3$ .
3
Two amplitudes have scale values 4 and 6 under $M = {log}_{10} (A / A_{0})$ . Find their larger-to-smaller ratio.
2 marks
$10^{6 - 4} = 100$ .
100
For $p = - {log}_{10} c$ , find $c$ when $p = 5$ .
2 marks
${log}_{10} c = - 5$ , so $c = 10^{- 5}$ .
$c = 10^{- 5}$
Why can zero not appear as a value on an ordinary real logarithmic axis?
3 marks
No real exponent of a positive valid base gives zero.
Its logarithm is undefined.

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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.

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