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Logarithms

§ Algebra

Logarithms

CCSS.HSF.BFCCSS.HSF.LE3 min read

A logarithm is the inverse operation of exponentiation, answering the question: to what power must a base be raised to produce a given number? The expression log₅(25) = 2 means that 5² = 25. Logarithms appear throughout Year 12 and 13 A-level mathematics, forming the foundation for exponential equations and growth models.

§ 01

Why it matters

Logarithms model exponential phenomena across science and finance. The Richter scale uses base-10 logarithms to measure earthquake intensity — each whole number represents a 10-fold increase in magnitude. In chemistry, pH measures acidity logarithmically, where pH 3 is 10 times more acidic than pH 4. Computer scientists use base-2 logarithms in algorithms and data structures, determining that a binary search through 1,024 items requires at most 10 steps. Financial analysts apply natural logarithms to calculate compound interest and investment growth rates. Population growth, radioactive decay, and sound intensity all follow logarithmic patterns that require these mathematical tools for accurate modelling and prediction.

§ 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_5(25) = _______

Answer: 2

  1. Understand what a logarithm asks → log_5(25) = ? means: 5^? = 25 — A logarithm answers the question: '5 raised to WHAT power gives 25?'
  2. Try powers of 5 → 51 = 5, 52 = 25 — Calculate 5^1, 5^2, ... until we reach 25.
  3. Read off the exponent → 52 = 25, so log_5(25) = 2 — The exponent that gives 25 is 2. That's our answer.
Easy§ 02

log_5(25) = _______

Answer: 2

  1. Rewrite as an exponential equation → log_5(25) = n means 5n = 25 — Converting between log form and exponential form is the key skill.
  2. Build up powers of 5 → 51 = 5, 52 = 25 — Calculate successive powers of 5 until we hit 25.
  3. Identify the matching power → 52 = 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(103) = _______

Answer: 3

  1. Recall the power rule for logarithms → log(an) = n · log(a) — The exponent comes out as a multiplier. This is the third main log rule.
  2. Apply the rule → log_10(103) = 3 · log_10(10) — Move the exponent 3 in front of the log.
  3. Evaluate log_10(10) → log_10(10) = 1 (since 101 = 10) — 10 raised to 1 gives 10.
  4. Multiply → 3 × 1 = 3 — Multiply the exponent by the log value.
§ 04

Common mistakes

  • Confusing the base and argument positions, writing log₂(8) = 2 instead of 3 when 2³ = 8.
  • Incorrectly applying the product rule as log(3 × 4) = log(3) × log(4) instead of log(3) + log(4).
  • Writing log₅(25) + log₅(5) = log₅(30) instead of log₅(125) when using the product rule backwards.
§ 05

Frequently asked questions

What is the difference between log and ln?
log typically refers to the common logarithm (base 10), whilst ln represents the natural logarithm (base e ≈ 2.718). Both follow identical rules but have different bases. Natural logarithms appear frequently in calculus and exponential growth models, whilst common logarithms are used in scientific calculations and the Richter scale.
How do you convert between different logarithm bases?
Use the change of base formula: log_a(x) = log_b(x) ÷ log_b(a). For example, log₂(8) = log₁₀(8) ÷ log₁₀(2) = 0.903 ÷ 0.301 = 3. This allows calculation of any logarithm using a calculator that only has log₁₀ or ln buttons.
Why does log₁₀(1) equal zero for any base?
Any number raised to the power of zero equals 1, so log_b(1) = 0 for any valid base b. This means 10⁰ = 1, 2⁰ = 1, and 5⁰ = 1. The logarithm asks 'what power gives 1?' and the answer is always zero, regardless of the base used.
How do you solve exponential equations using logarithms?
Take the logarithm of both sides, then use logarithm rules to isolate the variable. For 2ˣ = 16, take log₂ of both sides: log₂(2ˣ) = log₂(16). The left side simplifies to x, and log₂(16) = 4, so x = 4. This method works for any exponential equation.
What logarithm laws do GCSE students need to know?
Three fundamental laws: product rule log(ab) = log(a) + log(b), quotient rule log(a/b) = log(a) - log(b), and power rule log(aⁿ) = n·log(a). These appear in Year 12 specifications and are essential for solving exponential equations and simplifying logarithmic expressions in A-level mathematics.
§ 06

See also

LogarithmBaseExpressionVariableEquation
§ 06

Where to next?

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