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Math · Functions

Logarithms

Convert between exponential and logarithmic form, and apply log rules to simplify and solve — a logarithm is just an exponent asking a question in a different order. Every question comes with a written walkthrough of the rule applied.

Practise logarithms nowSee sample questions
3
Difficulty tiers
35s
Target pace
50%
Average first-attempt score

What's covered

  • Converting between forms — log_b(x) = y means the same thing as b^y = x — rewriting between the two forms is the first move on almost every problem.
  • The product and quotient rules — log(ab) = log(a) + log(b), and log(a/b) = log(a) − log(b) — logs turn multiplication into addition and division into subtraction.
  • The power rule — log(aⁿ) = n·log(a) — an exponent inside a log becomes a coefficient outside it.
  • Common and natural logs — log without a written base means base 10; ln means base e — both follow the same rules as any other base.
  • Solving exponential equations with logs — take the log of both sides to bring a variable down from an exponent when it can't be isolated any other way.

Where students lose marks

Treating log(a+b) as log(a)+log(b)

The product rule applies to log(a×b), not log(a+b) — there's no rule that simplifies the log of a sum at all.

Mixing up log_b(1) and log_b(b)

log_b(1) = 0 and log_b(b) = 1 for any valid base b — these two facts get swapped under time pressure more than almost any other rule in this subtopic.

Applying the power rule to the wrong part of the expression

In (log a)ⁿ, the exponent applies to the whole logarithm, not to a — that's a completely different expression from log(aⁿ), where the power rule actually applies.

Three sample questions

Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.

Sample 01Foundation

Write 3⁴ = 81 in logarithmic form.

Sample 02Core

Solve for x: log₂(x) + log₂(5) = log₂(40)

Sample 03Advanced

Solve for x: 5^(2x+1) = 15

How to practise this

Logarithms feel abstract until the exponential-form conversion becomes automatic — after that, most problems are just the product, quotient, and power rules in different combinations.

Rewrite in the other form first

Before applying any rule, convert between exponential and logarithmic form if the problem doesn't already give you the form you want. Untimed practice is where that conversion becomes reflexive.

Then 35 seconds a question

Move to timed sessions once rule selection is fast. Equations that require taking the log of both sides run a bit longer than direct evaluation.

Fold into a mock

Pair logarithms with exponents in a mock — they're the same relationship read in two directions.

Stop fearing the notation.
Start converting the form.

  • A rule-by-rule walkthrough on every item
  • Timed or untimed sessions, any length you like
  • Tracked separately — see your logarithms accuracy and pace over time
Start practising free

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Evaluating and graphing functionsLinear functionsQuadratic functionsAll of Math