Common Mistakes in IGCSE Add Maths Logarithms

Q: What are the log laws in IGCSE additional mathematics?

The Cambridge IGCSE Additional Mathematics (0606) syllabus requires students to know: log(ab) equals log a plus log b; log(a/b) equals log a minus log b; log(a to the power n) equals n times log a; and the change-of-base formula log base b of a equals (log a) divided by (log b). These four laws are the basis of all logarithm questions in the 0606 exam.

Q: Is ln in IGCSE additional mathematics?

Yes. The natural logarithm (ln, or log base e) is part of the Cambridge IGCSE Additional Mathematics (0606) syllabus. Students need to know that ln x means log base e of x, that e is approximately 2.718, and that ln(e to the power x) equals x. Confusing ln with log base 10 is a common error in differentiation and integration questions.

Q: How do you solve exponential equations in IGCSE add maths?

To solve an exponential equation in IGCSE Additional Mathematics (0606): take the log of both sides (using ln or log base 10), apply the power law to bring the exponent down as a multiplier, then solve the resulting linear or quadratic equation. The most common mark-loss points are: forgetting to take the log of both sides completely, and not checking that the solution gives a positive argument inside the log.

Q: Why do students find logarithms hard in IGCSE Add Maths?

Students find logarithms hard in IGCSE Additional Mathematics because they memorise the three log laws as isolated facts rather than understanding what they represent. When a question applies the laws in a non-standard order (for example, a log equation embedded inside a quadratic), students cannot adapt. Understanding why each law holds — as the inverse of index laws — makes non-standard questions predictable.

Short answer

What are the most common mistakes in IGCSE Add Maths logarithms?

The five most common IGCSE Additional Mathematics logarithm mistakes are: applying log laws in the wrong direction, taking the log of a negative number or zero, confusing log base 10 with natural log (ln), forgetting to apply the change-of-base formula when needed, and not checking whether the solution satisfies the domain restriction. Each is a process error, not a concept gap.

Mark analysis

Why logarithms is a consistent mark-loss topic in IGCSE Add Maths

Logarithmic and exponential functions (Chapter 7 of the Cambridge IGCSE Additional Mathematics 0606 syllabus) appears across both papers, often combined with other topics — a log equation that simplifies to a quadratic, or an exponential equation that requires differentiation. Students who understand the log laws in isolation often lose marks when the topic appears in an unfamiliar combination.

Mistake 1

Log laws applied in the wrong direction

The law log(ab) equals log a plus log b means that multiplication inside the log becomes addition outside. The reverse is also true: log a plus log b can be combined into log(ab). Students often mix up which direction to apply the law — expanding when they should combine, or combining when they should expand.

Where it goes wrong. When a question asks to simplify log(2x) plus log(3x), many students write log(2x) plus log(3x) equals log(2x times 3x) equals log(6x squared) correctly. But when the question writes it as log(6x squared) and asks to expand, they write log 6 plus log x instead of log 6 plus 2 log x, missing the power law application.
The fix. Before combining or expanding, write the law you are using on the same line. This forces you to apply it deliberately rather than from memory.
Exam pattern. Log simplification questions often chain two or three law applications. A wrong first step means every subsequent step is wrong, even if the method is correct.

Mistake 2

Taking the log of a negative number or zero

The logarithm of a negative number is undefined in real numbers. When solving a logarithmic equation, the solution must produce a positive argument inside the log. Students who find two solutions algebraically sometimes accept a negative solution without checking.

Where it goes wrong. When a log equation simplifies to a quadratic, both roots of the quadratic must be checked against the original equation. If one root gives log(negative number), it is rejected. Students who forget this step write both roots as the answer.
The fix. After finding solutions to a log equation, substitute each one back into the original expression inside the log. If the argument is zero or negative, reject that solution and state why.
Exam mark implication. The mark scheme usually awards one mark for the correct method and one mark for the correct accepted solution(s). Accepting an invalid solution loses the second mark even when the method is correct.

Mistake 3

Confusing log base 10 with natural log (ln)

In IGCSE Additional Mathematics, "log" means log base 10 and "ln" means log base e (natural log). These two functions are not the same, and applying one where the other is required gives a wrong numerical answer.

Where it goes wrong. When differentiating or integrating, the rules involve ln (natural log), not log base 10. Students who use log base 10 in differentiation questions lose all method marks because the derivative of log base 10 of x is 1 divided by (x times ln 10), not simply 1 divided by x.
The fix. In any calculus question involving logarithms, the log must be ln (natural log). Read the question: if it writes log, it means base 10; if it writes ln, it means base e. In differentiation and integration, the base is always e.
Linked mistake. Students sometimes write ln e equals 1 correctly but then write e to the power ln x incorrectly as e times x rather than x. The inverse relationship (e to the power ln x equals x) is worth drilling separately.

Mistake 4

Forgetting the change-of-base formula

When a logarithm has a base other than 10 or e, a calculator cannot evaluate it directly. The change-of-base formula converts it: log base b of a equals (log a) divided by (log b). Students who do not recognise when this formula is needed cannot evaluate or simplify non-standard bases.

Where it goes wrong. A question might ask to solve log base 3 of x equals 2.5. Students who try to evaluate this directly on a calculator without applying change-of-base get a wrong answer or give up. The correct route is: x equals 3 to the power 2.5, which does not require change of base — but a question asking for log base 3 of 20 does.
The fix. When you see a logarithm with a base that is not 10 or e and the question asks for a numerical value, reach for the change-of-base formula first. Write it out as (log 20) divided by (log 3) and evaluate.
Exam context. Change-of-base questions in 0606 often appear as multi-part questions where part (a) derives a log expression and part (b) asks for a numerical answer. Students who cannot apply change-of-base lose part (b) even when part (a) is correct.

Mistake 5

Not checking the solution satisfies the domain restriction

Every logarithmic expression has a domain restriction: the argument must be strictly positive. When a log equation produces two solutions, both must be checked against this restriction. Students who skip the check accept invalid solutions.

Where it goes wrong. When an equation such as log(x squared minus 4) equals 0 is solved, the algebraic solutions are x equals plus or minus the square root of 5. Both must be substituted into the original expression to check the argument is positive. In this case, both solutions give a positive argument and are valid. But when one gives a negative argument, students who do not check accept the wrong solution.
The fix. After solving, always write a check step: "Check x equals [value]: argument = [positive number], valid" and "Check x equals [other value]: argument = [negative number], rejected." The check takes four lines and prevents a full mark loss.
The weak step underneath. Students who skip this step usually do so because they were not taught domain restrictions as part of the log chapter — they learned the laws but not the constraints. Understanding why the argument must be positive (the log of zero or a negative is undefined in real numbers) makes the check feel logical rather than a box-ticking exercise.

What this means for your child

These are process errors, not concept gaps

Every mistake above is about missing a step in a written process, not about failing to understand logarithms. Students who make these errors typically know what log laws are. The problem is that they learned the laws without building the habit of writing out each step and checking.

In class, Teacher Au addresses this by working through the same question twice: once at the student's current speed (where the error appears) and once with every intermediate step written down (where the error disappears). For logarithms specifically, Teacher Au builds the connection between log laws and index laws early in the chapter so students understand why each law holds rather than memorising it as a formula.

The goal is exam consistency: a student who checks the domain restriction on every log question will never lose that mark, and a student who understands log as the inverse of indices will not be caught out by non-standard applications.

What is the hardest topic in IGCSE Add Maths?

Where logarithms sits in the full difficulty ranking and how it connects to calculus.

Read the topic difficulty guide

Common mistakes in IGCSE Add Maths calculus

The same pattern of process errors in differentiation and integration — including ln in calculus questions.

Read the calculus mistakes guide

Questions parents ask

Frequently asked questions

What are the most common mistakes in IGCSE Add Maths logarithms?

The five most common IGCSE Additional Mathematics logarithm mistakes are: applying log laws in the wrong direction, taking the log of a negative number or zero, confusing log base 10 with natural log (ln), forgetting to apply the change-of-base formula when needed, and not checking whether the solution satisfies the domain restriction. Each is a process error, not a concept gap.

What are the log laws in IGCSE additional mathematics?

Is ln in IGCSE additional mathematics?

How do you solve exponential equations in IGCSE add maths?

Why do students find logarithms hard in IGCSE Add Maths?

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