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§ Algebra

Logarithms

CCSS.HSF.BFCCSS.HSF.LE3 min read

Logarithms bridge the gap between GCSE algebra and A-level mathematics, appearing in Year 12 curricula across England. These inverse operations to exponentiation unlock exponential equations that standard algebraic methods cannot solve.

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§ 01

Why it matters

Logarithms model real-world phenomena with exponential patterns. Sound engineers use decibel scales (logarithmic) where 60 dB is 10 times louder than 50 dB. Population growth models rely on natural logarithms — if bacteria double every 20 minutes, logarithms calculate when 1,000 bacteria become 64,000. Financial compound interest calculations use logarithms to determine investment timescales. Earthquake measurements on the Richter scale are logarithmic, where magnitude 7 represents 10 times more energy than magnitude 6. Computer science algorithms often have logarithmic time complexity, making them efficient for large datasets. Chemistry uses pH scales (logarithmic) where each unit represents a 10-fold change in acidity. Students encounter logarithms in GCSE Higher tier and extensively in A-level Core Mathematics, making mastery essential for university STEM courses.

§ 02

How to solve logarithms

Logarithms

  • log_b(x) = n means bn = x.
  • Product: log(ab) = log(a) + log(b).
  • Quotient: log(a/b) = log(a) − log(b).
  • Power: log(an) = n·log(a).

Example: log₂(8) = 3 because 2³ = 8.

§ 03

Worked examples

Beginner§ 01

log_3(27) = _______

Answer: 3

  1. Understand what a logarithm asks → log_3(27) = ? means: 3^? = 27 — A logarithm answers the question: '3 raised to WHAT power gives 27?'
  2. Try powers of 3 → 3^1 = 3, 3^2 = 9, 3^3 = 27 — Calculate 3^1, 3^2, ... until we reach 27.
  3. Read off the exponent → 3^3 = 27, so log_3(27) = 3 — The exponent that gives 27 is 3. That's our answer.
Easy§ 02

log_5(25) = _______

Answer: 2

  1. Rewrite as an exponential equation → log_5(25) = n means 5^n = 25 — Converting between log form and exponential form is the key skill.
  2. Build up powers of 5 → 5^1 = 5, 5^2 = 25 — Calculate successive powers of 5 until we hit 25.
  3. Identify the matching power → 5^2 = 25 ← match! — The 2th power of 5 equals 25.
  4. Write the answer → log_5(25) = 2 — The logarithm equals the exponent.
Medium§ 03

log_10(102) = _______

Answer: 2

  1. Recall the power rule for logarithms → log(a^n) = n · log(a) — The exponent comes out as a multiplier. This is the third main log rule.
  2. Apply the rule → log_10(10^2) = 2 · log_10(10) — Move the exponent 2 in front of the log.
  3. Evaluate log_10(10) → log_10(10) = 1 (since 10^1 = 10) — 10 raised to 1 gives 10.
  4. Multiply → 2 × 1 = 2 — Multiply the exponent by the log value.
§ 04

Common mistakes

  • Students often confuse logarithm direction, writing log₂(8) = 2 instead of 3, forgetting that 2³ = 8, not 2² = 8
  • When applying the product rule, students incorrectly write log(2 × 4) = log(2) × log(4) instead of log(2) + log(4) = log(8)
  • Students frequently misapply the power rule, writing log(5²) = (log 5)² instead of 2 × log(5), confusing multiplication with exponentiation
  • Converting between exponential and logarithmic form causes errors — students write 3⁴ = 81 as log₈₁(3) = 4 instead of log₃(81) = 4
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§ 05

Frequently asked questions

Why do we need logarithms when we have calculators?
Logarithms solve exponential equations that standard algebra cannot handle. When solving 2ˣ = 50, algebraic manipulation fails, but logarithms provide the exact solution x = log₂(50). They're essential for modelling exponential growth, radioactive decay, and compound interest problems in A-level mathematics.
What's the difference between common and natural logarithms?
Common logarithms use base 10 (written log or log₁₀), whilst natural logarithms use base e ≈ 2.718 (written ln). Natural logarithms appear frequently in calculus and exponential growth models. Both follow identical rules, but natural logarithms connect directly to continuous growth rates in advanced mathematics.
How do I remember the logarithm rules?
Think of logarithms as 'undoing' exponentiation. The product rule log(ab) = log(a) + log(b) mirrors how multiplication becomes addition in exponents: x^a × x^b = x^(a+b). Similarly, division becomes subtraction, and powers become multiplication. This connection helps students remember all three fundamental logarithm laws.
When do logarithms appear in GCSE versus A-level?
GCSE Higher tier introduces basic exponential equations, but logarithms feature prominently in Year 12 A-level Core Mathematics. Students learn logarithm laws, solve exponential equations, and apply them to growth/decay models. Year 13 extends to natural logarithms and complex exponential functions throughout further pure mathematics.
Why can't we take logarithms of negative numbers?
Since logarithms ask 'what power gives this result?', and positive bases raised to any real power always produce positive results, negative inputs are undefined in real numbers. For example, no real power makes 10ˣ = -5. Complex logarithms exist, but Year 12 students work exclusively with positive arguments.
§ 06

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