Interpreting logarithmic scales | Year 10

01 / LEARN

Understand the idea.

Equal spacing need not mean equal addition

On a base- $10$ logarithmic axis, position is determined by ${log}_{10}$ of the quantity. Labels $1, 10, 100, 1000$ are equally spaced because their logarithms are $0, 1, 2, 3$ , rather than because their differences are equal.

A displacement of $d$ logarithmic units represents a multiplicative ratio $10^{d}$ . This displays very different orders of magnitude on one axis. Always read whether an axis is linear or logarithmic before comparing distances.

Use the stated definition

Use the exact definition of the scale. If $M = {log}_{10} (\frac{A}{A_{0}})$ , subtracting two scale values gives ${log}_{10} (\frac{A_{2}}{A_{1}})$ , so a difference $d$ corresponds to amplitude ratio $10^{d}$ , not an energy ratio.

For the supplied classroom model $p = - {log}_{10} c$ with $c > 0$ , the minus sign reverses the direction: increasing $p$ by $1$ divides $c$ by $10$ . Do not drop the minus sign when evaluating or interpreting this model.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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?

Worked solution

1
Form a ratio of the quantities.
$ratio = \frac{10^{5}}{10^{2}}$
How many times larger asks for division, not subtraction.
2
Subtract the exponents.
$\frac{10^{5}}{10^{2}} = 10^{5 - 2} = 10^{3}$
Division of powers with the same nonzero base subtracts indices.
3
Interpret the ratio.
$10^{3} = 1000$
A difference of three exponent units means the second quantity is $1000$ times the first.

Answer

1000 times

EXAMPLE 02

Use a supplied formula

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

Worked solution

1
Substitute the supplied concentration.
$p = - {log}_{10} (10^{- 4})$
The logarithm input is positive, so the real expression is defined.
2
Evaluate the logarithm.
${log}_{10} (10^{- 4}) = - 4$
A base- $10$ logarithm recovers the exponent.
3
Apply the outside negative sign.
$p = - (- 4) = 4$
The supplied formula negates the logarithm value.

Answer

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$10^{1.5} \approx 31.6$

A difference of $1.5$ logarithmic units means a ratio $10^{1.5} = 10 \sqrt{10} \approx 31.6$ , rather than a difference of $1.5$ in the original quantities.

Keep practising

6 worksheet questions · 13 marks

Open worksheet 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 $M = {log}_{10} (\frac{A}{A_{0}})$ , find $M$ when $A = 1000 A_{0}$ .
2 marks
Two amplitudes have scale values 4 and 6 under $M = {log}_{10} (\frac{A}{A_{0}})$ . 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 $\frac{1 + 3}{2} = 2$ , 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 $M = {log}_{10} (\frac{A}{A_{0}})$ , find $M$ when $A = 1000 A_{0}$ .
2 marks
$M = {log}_{10} 1000 = 3$ .
3
Two amplitudes have scale values 4 and 6 under $M = {log}_{10} (\frac{A}{A_{0}})$ . 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.

Official curriculum source ↗

In this lesson

Understand the idea.

Equal spacing need not mean equal addition

Use the stated definition

See the method in action.

Read a ratio

Form a ratio of the quantities.

Subtract the exponents.

Interpret the ratio.

Use a supplied formula

Substitute the supplied concentration.

Evaluate the logarithm.

Apply the outside negative sign.

Try it yourself.

Check your understanding

Keep practising