Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 3 - Logarithmic scales in applied contexts

Logarithmic scales in applied contexts

Interpret base-10 logarithmic scales and compare multiplicative changes represented by equal scale intervals.

Updated 2026-07-26 - 12 min read

Logarithmic scales in applied contexts is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Interpret logarithmic scale steps

On a base-10 logarithmic scale, an increase of one scale unit represents multiplication of the underlying quantity by 10. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by interpret logarithmic scale steps and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Equal scale intervals represent equal ratios, not equal absolute differences.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

A base-10 scale rises from 3 to 5. By what factor does the underlying quantity change?

Step 1 - identify the governing idea: On a base-10 logarithmic scale, an increase of one scale unit represents multiplication of the underlying quantity by 10.

Step 2 - apply it to this evidence: Two scale units represent 10² = 100.

Result: 100

The relationship is visible in the working: Two scale units represent 10² = 100. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 2 — It does not agree with the required relationship: Two scale units represent 10² = 100.
  • 20 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1000 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

What factor corresponds to a decrease of 3 on a base-10 logarithmic scale?

Step 1 - identify the governing idea: On a base-10 logarithmic scale, an increase of one scale unit represents multiplication of the underlying quantity by 10.

Step 2 - apply it to this evidence: A decrease of 3 multiplies by 10⁻³.

Result: 1/1000

The relationship is visible in the working: A decrease of 3 multiplies by 10⁻³. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 1/3 — It does not agree with the required relationship: A decrease of 3 multiplies by 10⁻³.
  • −30 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1000 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Two values differ by 0 on the same log scale. What is their underlying ratio?

Step 1 - identify the governing idea: On a base-10 logarithmic scale, an increase of one scale unit represents multiplication of the underlying quantity by 10.

Step 2 - apply it to this evidence: 10⁰ = 1, so the values are equal.

Result: 1

The relationship is visible in the working: 10⁰ = 1, so the values are equal. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 0 — It does not agree with the required relationship: 10⁰ = 1, so the values are equal.
  • 10 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Undefined — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Compare logarithmic measurements

Subtract scale readings first, then convert the difference to a multiplicative ratio using the scale base. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by compare logarithmic measurements and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Ratios come from scale differences: for base 10, a difference of 3 gives factor 1000.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

Measurements are 4.2 and 6.2 on a base-10 scale. Find the ratio.

Step 1 - identify the governing idea: Subtract scale readings first, then convert the difference to a multiplicative ratio using the scale base.

Step 2 - apply it to this evidence: The difference is 2, so the ratio is 10².

Result: 100

The relationship is visible in the working: The difference is 2, so the ratio is 10². Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 2 — It does not agree with the required relationship: The difference is 2, so the ratio is 10².
  • 10 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 6.2/4.2 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

One event has underlying intensity 1000 times another. What is the scale difference?

Step 1 - identify the governing idea: Subtract scale readings first, then convert the difference to a multiplicative ratio using the scale base.

Step 2 - apply it to this evidence: 1000 = 10³.

Result: 3

The relationship is visible in the working: 1000 = 10³. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 1000 — It does not agree with the required relationship: 1000 = 10³.
  • 10 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1/3 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

A reading drops from 7.5 to 6.5. What happens to the underlying quantity?

Step 1 - identify the governing idea: Subtract scale readings first, then convert the difference to a multiplicative ratio using the scale base.

Step 2 - apply it to this evidence: A drop of one unit gives factor 10⁻¹.

Result: It becomes one tenth

The relationship is visible in the working: A drop of one unit gives factor 10⁻¹. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • It drops by one unit only — It does not agree with the required relationship: A drop of one unit gives factor 10⁻¹.
  • It halves — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • It becomes zero — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Reason about logarithmic-scale contexts

State the scale base and what physical quantity the logarithm represents before interpreting a comparison. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by reason about logarithmic-scale contexts and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Different contexts may use different reference quantities, factors or definitions even when both are logarithmic.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

Why can equal visual gaps on a log graph represent very different absolute changes?

Step 1 - identify the governing idea: State the scale base and what physical quantity the logarithm represents before interpreting a comparison.

Step 2 - apply it to this evidence: Equal position steps multiply values, so absolute differences grow with magnitude.

Result: The axis encodes ratios

The relationship is visible in the working: Equal position steps multiply values, so absolute differences grow with magnitude. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • The graph is inaccurate — It does not agree with the required relationship: Equal position steps multiply values, so absolute differences grow with magnitude.
  • Values are rounded to integers — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The axis has no units — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

What must accompany a claim that one reading is '10 times larger'?

Step 1 - identify the governing idea: State the scale base and what physical quantity the logarithm represents before interpreting a comparison.

Step 2 - apply it to this evidence: The interpretation depends on what the logarithm encodes.

Result: The scale definition and underlying quantity

The relationship is visible in the working: The interpretation depends on what the logarithm encodes. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Only the graph colour — It does not agree with the required relationship: The interpretation depends on what the logarithm encodes.
  • No supporting information — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • A linear trend line — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Why is zero on a logarithmic scale not necessarily zero underlying quantity?

Step 1 - identify the governing idea: State the scale base and what physical quantity the logarithm represents before interpreting a comparison.

Step 2 - apply it to this evidence: A log value of zero corresponds to ratio 1.

Result: It can represent the reference quantity because log(1)=0

The relationship is visible in the working: A log value of zero corresponds to ratio 1. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Log scales invent data — It does not agree with the required relationship: A log value of zero corresponds to ratio 1.
  • Zero always means nothing — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Negative readings are impossible — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. What factor corresponds to a decrease of 3 on a base-10 logarithmic scale?
  2. One event has underlying intensity 1000 times another. What is the scale difference?
  3. What must accompany a claim that one reading is '10 times larger'?

Answers

  1. 1/1000 — A decrease of 3 multiplies by 10⁻³.
  2. 3 — 1000 = 10³.
  3. The scale definition and underlying quantity — The interpretation depends on what the logarithm encodes.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources