← All posts

Logarithms Explained Simply: Log Rules and Worked Examples

May 9, 2026 · 7 min · logarithms · log rules · algebra · a-level maths · exam technique

Written & checked by Rabail, a student.

Quick answer: A logarithm asks one question: what power do I raise the base to in order to get this number? So log base 10 of 1000 = 3, because 10^3 = 1000. Written generally, if a^x = b then log base a of b = x. Every log rule and every log equation in your exam is that one question rearranged.

I lost about three weeks to logarithms. My problem was that I kept reading "log x" as though log were a number multiplied by x, so I tried to cancel it, divide by it, all the wrong instincts. It is not a number, it is a question. My teacher made me say "what power?" out loud before writing anything, which felt ridiculous, but it worked. Here is the version I wish someone had given me on day one.

What a logarithm actually is

A logarithm is the reverse of raising to a power. Powers take a base and an exponent and give an answer: 2^5 = 32. Logs take the base and the answer and give back the exponent: log base 2 of 32 = 5.

The two forms carry identical information:

  • Index form: a^x = b
  • Log form: log base a of b = x

Read the log form as "the power you put on a to get b". Say it in words every time until it stops feeling strange:

  • 10^2 = 100 becomes log base 10 of 100 = 2
  • 5^3 = 125 becomes log base 5 of 125 = 3
  • 2^(-3) = 1/8 becomes log base 2 of (1/8) = -3

That last one is worth staring at. Logs of numbers between 0 and 1 are negative, because it takes a negative power to shrink a base below 1.

Two shorthands: a log with no base, like log(50), means base 10 in most school syllabuses, and ln(50) means base e, about 2.718 — the natural log button on your calculator.

Swapping between the two forms is the whole skill

Most log questions are solved the moment you rewrite them in the other form. Drill this before you touch the rules.

Solve log base 4 of x = 3. Swap to index form: x = 4^3 = 64. One line.

Solve log base x of 81 = 2. Swap: x^2 = 81, so x = 9. Note I did not write x = -9, because a log base must be positive and cannot be 1.

Solve log base 3 of 81 = x. Swap: 3^x = 81, and since 81 = 3^4, x = 4.

Same skill, three different unknowns. When you stall in an exam, the first thing to try is always the swap. I keep both forms on one card in my flashcards deck and it is still the card I review most.

The three log rules and what they are not

There are three rules, they all come from index laws, and they all have the same base throughout:

  • Product rule: log(a × b) = log(a) + log(b)
  • Quotient rule: log(a / b) = log(a) - log(b)
  • Power rule: log(a^n) = n × log(a)

Why the product rule works: multiplying powers adds the exponents, so 2^3 × 2^4 = 2^7. Logs give you exponents, so multiplying inside a log becomes adding outside it. Understand that sentence and you never have to memorise which rule uses plus and which uses minus.

Now the mistakes. These three cost more marks than anything else here:

  1. log(a + b) is NOT log(a) + log(b). The product rule needs multiplication inside the bracket. Adding inside a log cannot be split at all.
  2. log(a) / log(b) is NOT log(a/b). Dividing two separate logs is a different object from the log of a fraction.
  3. (log a)^2 is NOT 2 log(a). The power rule applies to a power inside the bracket, not to the whole log squared.

Also useful as a final step: log base a of a = 1, and log base a of 1 = 0.

Worked example: solving 3^x = 40

This is the classic "unknown in the exponent" question, and it appears in Cambridge Additional Maths, A-Level, AP and CBSE Class 11 papers in almost identical wording.

  1. Start: 3^x = 40. You cannot write 40 as a neat power of 3, so index matching will not work.
  2. Take log base 10 of both sides: log(3^x) = log(40).
  3. Apply the power rule to bring x down: x log(3) = log(40).
  4. Divide: x = log(40) / log(3).
  5. Calculator: log(40) = 1.60206, log(3) = 0.47712, so x = 1.60206 / 0.47712 = 3.3578.
  6. Answer: x = 3.36 to 3 significant figures.

Two examiner points. Step 3 is where the method marks live, so write it out even if your calculator has a log-base button — a bare final answer usually scores 1 of 3. And do not round log(3) to 0.48 mid-question; rounding early shifts the answer and loses the accuracy mark.

The same structure handles change of base: log base 2 of 50 = log(50) / log(2) = 1.69897 / 0.30103 = 5.64 to 3 significant figures.

Worked example: a log equation with a hidden trap

Solve log(x) + log(x - 3) = 1, where log means base 10.

  1. Combine using the product rule: log(x(x - 3)) = 1.
  2. Swap to index form: x(x - 3) = 10^1 = 10.
  3. Expand and rearrange: x^2 - 3x - 10 = 0.
  4. Factorise: (x - 5)(x + 2) = 0, so x = 5 or x = -2.
  5. Check the domain. You cannot take the log of zero or a negative number, and the original equation contains log(x - 3), so you need x > 3. That kills x = -2 immediately.
  6. Answer: x = 5 only.

Step 5 is the point of the question. Mark schemes routinely award a mark for rejecting the invalid root, and plenty of students hand over both answers and lose it. Write the reason: "x = -2 rejected as log(-2) is undefined." To check your working line by line, paste the equation into the math solver and compare its steps to yours, not just the final number.

Where logs actually show up

Logs are not on the standard GCSE maths papers for AQA, Edexcel or OCR, so if you are in Year 11 and panicking, you probably do not need this yet. They appear in Cambridge IGCSE Additional Maths, A-Level and International A-Level, CBSE Class 11 and 12, AP courses, and WASSCE maths.

Beyond pure maths, they rescue awkward equations. Half-life problems need a log to solve for time. The pH scale is a negative log of hydrogen ion concentration. Compound interest questions asking "after how many years" are exponent-unknown problems in disguise. That is the honest reason the topic exists: logs get a variable out of an exponent. More practice sits on the maths hub, and if one step will not click, ask for it broken down in explain.

Test yourself

  1. Write 5^4 = 625 in logarithm form.
  2. Simplify log(8) + log(5) - log(4) into a single logarithm, then evaluate it in base 10.
  3. Solve 2^x = 90, giving your answer to 3 significant figures.

Answers: log base 5 of 625 = 4; log(8 × 5 / 4) = log(10) = 1; x = log(90) / log(2) = 6.49.

FAQ

Do I need logarithms for GCSE maths?

No. Logs are not on the AQA, Edexcel or OCR GCSE maths specifications. You meet them at A-Level, in Cambridge IGCSE Additional Maths, in CBSE Class 11, in AP maths courses and in WASSCE maths. If you are doing GCSE and a log has appeared in your homework, it is enrichment, not exam content.

What is the difference between log and ln?

Same operation, different bases. Written plainly, log usually means base 10 and ln means base e, roughly 2.718. Both obey the identical three rules. If you are solving a^x = b it genuinely does not matter which you pick, because the base cancels in the division.

Why can you not take the log of a negative number or zero?

Because no real power produces them. Ask "what power of 10 gives -5?" — every real power of 10 is positive, so no answer exists. Same for zero: 10^x shrinks as x drops but never reaches 0. This is exactly why domain checks matter.

Should I memorise the log rules?

Yes, and test yourself on them rather than rereading them. Some formula booklets list them and some do not, and even when given, hunting for them mid-paper costs time. Three rules is a tiny memorisation cost. A quick self-test on the quiz tool checks they have stuck.

In short: a logarithm is a question about a power, not a number attached to one. Swap between a^x = b and log base a of b = x instantly, memorise the three rules, take logs of both sides whenever the unknown sits in an exponent, and check the domain before writing your final answer.