Year 10 - Logarithmic scales - Free practice

Year 10 Logarithmic Scales Worksheets & Practice

Read, interpret and use log scales in real contexts. Worked solutions every time, free.

Quick answer

What are logarithmic scales and how do you use them?

A logarithmic scale spaces numbers so each equal step represents multiplying by the same factor (often 10), not adding. To convert a log scale reading to the original quantity, calculate 10 to the power of that reading. For example, a pH of 3 means a hydrogen ion concentration of 10⁻³ mol/L, which is 10 times more acidic than pH 4.

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Worked examples

Year 10 logarithmic scales: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Converting a log scale reading to the original quantity
The pH scale is a base-10 logarithmic scale defined by $\text{pH} = -\log_{10}[\text{H}^+]$ , where $[\text{H}^+]$ is the hydrogen ion concentration in moles per litre. A solution has a pH of 3. Find its hydrogen ion concentration.
Show worked solution
Key idea: On a base-10 log scale, if the log scale reading is $L$ , the original quantity is $10^L$ . Here, $\text{pH} = -\log_{10}[\text{H}^+]$ , so rearrange to find $[\text{H}^+]$ .
1. Write the equation with the known pH: $3 = -\log_{10}[\text{H}^+]$
2. Multiply both sides by $-1$ : $\log_{10}[\text{H}^+] = -3$
3. Convert from logarithmic form to index form: $[\text{H}^+] = 10^{-3}$
4. Evaluate: $[\text{H}^+] = 0.001 \text{ mol/L}$
Answer: The hydrogen ion concentration is $0.001$ mol/L.
Example 2 · Interpreting a one-unit increase on a log scale
The Richter scale measures earthquake magnitude using a base-10 logarithmic scale. Earthquake A has magnitude 5 and Earthquake B has magnitude 7. How many times greater is the ground shaking intensity of Earthquake B than Earthquake A?
Show worked solution
Key idea: On a base-10 log scale, each one-unit increase in the reading corresponds to a tenfold (multiply by 10) increase in the actual quantity. A difference of $d$ units means the actual quantity increases by a factor of $10^d$ .
1. Find the difference in magnitudes: $7 - 5 = 2 \text{ units}$
2. Calculate the factor of increase: $10^2 = 100$
3. Interpret: Earthquake B has a ground shaking intensity 100 times greater than Earthquake A.
Answer: Earthquake B's intensity is $100$ times greater than Earthquake A's.
Example 3 · Choosing and applying a log scale in context
A sound's loudness in decibels is given by $L = 10\log_{10}\left(\dfrac{I}{I_0}\right)$ , where $I$ is the sound intensity in watts per square metre and $I_0 = 10^{-12}$ W/m² is the reference intensity. A lawnmower produces a sound level of 90 dB. A normal conversation produces a sound level of 60 dB. How many times more intense is the lawnmower than the normal conversation?
Show worked solution
Key idea: The decibel scale is base-10 logarithmic. A difference of $d$ decibels between two sounds corresponds to an intensity ratio of $10^{d/10}$ , because the formula uses $10\log_{10}$ , so dividing the decibel difference by 10 gives the exponent.
1. Find the difference in decibel levels: $90 - 60 = 30 \text{ dB}$
2. Divide by 10 to find the exponent (because the formula has the factor of 10 in front of the log): $\frac{30}{10} = 3$
3. Calculate the intensity ratio: $10^3 = 1000$
4. Interpret: The lawnmower is 1000 times more intense than a normal conversation.
Answer: The lawnmower is $1000$ times more intense.

Common mistakes

Where students usually lose marks

Adding instead of multiplying across log scale intervals

Moving 2 units up a base-10 log scale means the quantity increases by $2 \times 10 = 20$ .

Equal steps on a log scale represent equal multiplication, not equal addition. Moving 2 units up a base-10 log scale multiplies the quantity by $10^2 = 100$ , not by 20.

Forgetting the negative sign in the pH formula

$\text{pH} = 3 \Rightarrow \log_{10}[\text{H}^+] = 3 \Rightarrow [\text{H}^+] = 1000$ mol/L.

The pH formula is $\text{pH} = -\log_{10}[\text{H}^+]$ , so $\log_{10}[\text{H}^+] = -\text{pH}$ . Always apply the negative sign before converting to index form, giving $[\text{H}^+] = 10^{-3} = 0.001$ mol/L for pH 3.

Dividing the decibel difference by 10 when it is not needed

Two sounds differ by 20 dB, so the intensity ratio is $10^{20/10} = 10^2$ but then using the same step for the Richter scale: two earthquakes differ by 2 on the Richter scale, so the ratio is $10^{2/10}$ .

Only the decibel formula uses a factor of 10 in front of the log, so you divide the dB difference by 10 to get the exponent. The Richter scale uses $\log_{10}$ with no extra factor, so a difference of 2 gives a ratio of $10^2 = 100$ directly. Always check the formula for the scale you are using.

Thinking a log scale is useful for quantities of similar size

Plotting the heights of five students (between 150 cm and 190 cm) on a log scale to show the differences more clearly.

A log scale is most useful when the quantities span several orders of magnitude (very large range). When values are close together, a log scale compresses the differences and makes them harder to read. Use a linear scale for data with a small range.

Try one yourself

Your turn

The Richter scale measures earthquake magnitude logarithmically (base 10). An earthquake measures $6.5$ on the Richter scale. How many times greater is the ground shaking amplitude of this earthquake than one measuring $4.5$ ? Give your answer as a whole number.

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About this worksheet

Why we built it

Logarithmic scales turn enormous or tiny ranges of numbers into something readable on a single graph or chart. In Year 10, students learn to read values from log scales, compare quantities that span many powers of ten, and work with familiar applied contexts such as the Richter scale for earthquakes, the decibel scale for sound and the pH scale for acidity. These worksheets build that skill step by step, from understanding what each interval on a log scale actually represents through to solving problems that mix reading, interpreting and calculating.

What's covered

Sub-skills your student will practise

✓Reading values from a logarithmic scale on a graph or number line
✓Understanding that equal intervals represent multiplication, not addition
✓Comparing quantities using log scales (e.g. earthquake magnitudes, decibels, pH)
✓Converting between a log scale reading and the original quantity
✓Interpreting the effect of a one-unit increase on the actual quantity
✓Choosing when a log scale is more useful than a linear scale

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Frequently asked

Questions parents ask about logarithmic scales

What are logarithmic scales and why do Year 10 students study them?

A logarithmic scale spaces values by equal multiples rather than equal additions, so each step up represents ten times (or another fixed multiple) the previous value. Year 10 students study them because many real measurements, from earthquake strength to sound loudness, span ranges too wide to show usefully on a normal linear scale.

How do I help my child understand the difference between a log scale and a normal scale?

A helpful starting point is the Richter scale: a magnitude 6 earthquake is not twice as strong as a magnitude 3, it is one thousand times stronger. Practising with concrete examples like this, then moving to reading values off a printed log scale, builds the intuition quickly.

What kinds of real-world contexts appear in Year 10 log scale problems?

The Australian Curriculum emphasises applied contexts, so problems typically involve the Richter scale for earthquakes, decibels for sound intensity, pH for acids and bases, and sometimes astronomical distances or population data that spans many orders of magnitude.

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