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IB Maths: Laws of Logarithms & Exponents FAQ

Q: What are the most common exam mistakes with log and exponent laws?

The classic error is confusing $\log(ab) = \log a + \log b$ with $\log(a+b)$, which has no simple expansion. Students also mix up $a^m \times a^n = a^{m+n}$ with $(a^m)^n = a^{mn}$, and forget that $\log_a 1 = 0$ and $\log_a a = 1$ for any valid base.

Q: How do I know when to use the change of base formula?

Use change of base — $\log_a b = \dfrac{\log_c b}{\log_c a}$ — whenever your calculator only computes $\log_{10}$ or $\ln$, and you're asked for a log in another base, or need to compare logs with different bases in the same expression or equation.

Q: What are the laws of exponents I need for IB Maths?

You need: $a^m \times a^n = a^{m+n}$, $\dfrac{a^m}{a^n} = a^{m-n}$, $(a^m)^n = a^{mn}$, $a^0 = 1$ (for $a \neq 0$), $a^{-n} = \dfrac{1}{a^n}$, and $a^{1/n} = \sqrt[n]{a}$. These underpin exponential functions, compound interest and radioactive decay questions across both AA and AI.

Answered by RevisionPrep's IB Educators

Log and exponent laws look simple until a mixed-index equation or a change-of-base question turns up on Paper 1. Most marks lost here aren't about not knowing the rules — they're about applying them in the wrong order or forgetting a domain restriction. Here's what I see go wrong every year, and how to fix it.

Why students lose marks

Why do students lose marks on laws of logarithms & exponents in IB Maths?

Most marks are lost not from ignorance of the laws but from misapplying them — adding logs when they should multiply the arguments, or forgetting that $lo g (a + b) \neq = lo g a + lo g b$ . Domain errors (taking a log of a negative result) and skipping the change-of-base step also cost method marks routinely.

In fifteen years of marking mocks, the three repeat offenders are:

Sign errors under the log — solving $lo g_{2} (x - 3) = 2$ and forgetting to check $x - 3 > 0$ once you've found $x$ .
Distributing incorrectly — writing $lo g (x + y) = lo g x + lo g y$ , which is simply false.
Dropping the base — writing $lo g 8 = 3$ without stating base 2, which examiners will query on Paper 1 no-calculator questions.

Quick tip: after solving any log equation, always substitute back into the original equation — not the simplified one — to check the argument is positive.

What are the most common exam mistakes with log and exponent laws?

The classic error is confusing $lo g (ab) = lo g a + lo g b$ with $lo g (a + b)$ , which has no simple expansion. Students also mix up $a^{m} \times a^{n} = a^{m + n}$ with $(a^{m})^{n} = a^{mn}$ , and forget that $lo g_{a} 1 = 0$ and $lo g_{a} a = 1$ for any valid base.

A quick self-test before your next mock — can you evaluate these without a calculator?

$lo g_{5} 1$
$lo g_{3} 3$
$2^{0}$
$lo g_{2} \frac{1}{8}$

If any of those took more than five seconds, that's a fluency gap, not a conceptual one — drill it with short daily recall rather than long problem sets.

How do I know when to use the change of base formula?

Use change of base — $lo g_{a} b = \frac{lo g _{c} b}{lo g _{c} a}$ — whenever your calculator only computes $lo g_{10}$ or $ln$ , and you're asked for a log in another base, or need to compare logs with different bases in the same expression or equation.

Worked example: Evaluate $lo g_{5} 20$ using a GDC.

Rewrite as $\frac{ln 20}{ln 5}$ (or $\frac{lo g 20}{lo g 5}$ — either base works).
$ln 20 \approx 2.9957$ , $ln 5 \approx 1.6094$ .
Divide: $2.9957 \div 1.6094 \approx 1.861$ .

On Paper 2, examiners want to see the change-of-base line written out, not just a final decimal — that's where the method mark sits.

Core rules & how to study them

What are the laws of logarithms in IB Maths?

The core laws, given in the IB Maths AA and AI formula booklets, are: $lo g_{a} (x y) = lo g_{a} x + lo g_{a} y$ ; $lo g_{a} (\frac{x}{y}) = lo g_{a} x - lo g_{a} y$ ; $lo g_{a} (x^{n}) = n lo g_{a} x$ ; and $lo g_{a} a = 1$ , $lo g_{a} 1 = 0$ .

These all appear in Topic 1 (Number and Algebra) of the AA and AI guides, both examined from first assessment 2021 and still current for the 2025 syllabus. HL students additionally need the change-of-base formula and comfort manipulating logs inside exponential-growth models in Topic 3 (Functions) and Topic 5 (Calculus).

What are the laws of exponents I need for IB Maths?

You need: $a^{m} \times a^{n} = a^{m + n}$ , $\frac{a ^{m}}{a ^{n}} = a^{m - n}$ , $(a^{m})^{n} = a^{mn}$ , $a^{0} = 1$ (for $a \neq = 0$ ), $a^{- n} = \frac{1}{a ^{n}}$ , and $a^{1/ n} = n a$ . These underpin exponential functions, compound interest and radioactive decay questions across both AA and AI.

Common mistake: treating $a^{0} = 1$ as if it needs a calculation. It doesn't — it's a definition, and stating it without working is fine and expected in exam answers.

How do I get a 7 on log and exponent questions in IB Maths?

Getting full marks means treating logs and exponents as inverse operations you convert between fluently — not two separate topics. Practise rewriting exponential equations as logarithmic ones and back, show every law you apply by name in your working, and always state domain restrictions before finishing a solve.

A checklist I give my own students before a mock:

Can you solve $3^{x} = 20$ for $x$ using logs, showing the line $x = \frac{ln 20}{ln 3}$ ?
Can you simplify $lo g_{2} 8 + lo g_{2} 4$ without a calculator?
Do you state the domain restriction on any solved log equation?
Can you link exponential decay models (Topic 3) to log-linearisation for HL Paper 3-style questions?

If you can tick all four cleanly, the log/exponent section of Paper 1 should cost you nothing.

Are log and exponent questions harder on HL than SL?

Yes — HL extends the SL laws into solving more complex equations (systems with multiple exponential terms), applying logs to model real growth/decay data, and connecting them to calculus (differentiating $ln x$ and $e^{x}$ -based functions), which SL doesn't require.

SL students mostly need the laws for solving equations and simplifying expressions. HL adds log-linearisation of data (fitting $y = a b^{x}$ models), which shows up in both the AA and AI HL syllabuses and occasionally in HL Paper 3 investigative questions.

Syllabus & assessment specifics

Where do logs and exponents appear in the IB Maths syllabus?

According to the IB Mathematics: Analysis and Approaches and Mathematics: Applications and Interpretation guides, logarithm and exponent laws sit in Topic 1 (Number and Algebra), first examined 2021 and unchanged in structure for the current 2025 assessment cycle, with applications recurring in Topics 3 and 5.

Because this topic underpins exponential functions, compound growth, radioactive decay and even some statistics questions (log-likelihood in HL AI), it's genuinely cross-syllabus — a shaky grasp here costs marks well beyond the Topic 1 questions themselves.

Are logs and exponents allowed on the calculator paper or non-calculator paper?

Both. Paper 1 (no calculator, AA only) tests exact-value simplification using the laws by hand; Paper 2 and Paper 3 (calculator allowed) test applied problems — solving for unknowns in growth models, evaluating logs in unusual bases, and multi-step real-world contexts.

Quick tip: on Paper 1, if a question gives you "nice" numbers like $lo g_{3} 81$ or $2^{5} = 32$ , that's a strong signal you're expected to spot the exact value — reaching for decimal approximations there usually means you've missed the intended method.

Do I need logs and exponents for IB Maths Applications and Interpretation?

Yes — both AA and AI cover the same core log and exponent laws in Topic 1. AI leans more heavily on applied contexts (financial models, population growth, log-linear regression) while AA pushes further into algebraic manipulation and links to calculus.

Comparisons & choices

Is logs and exponents harder in AA or AI?

Neither is inherently harder — AA tests deeper algebraic manipulation of logs (proving identities, combining with calculus), while AI tests applied fluency (using logs in financial and growth models with a GDC). Students who prefer abstract algebra tend to find AA's treatment more natural; those who prefer applied number-crunching find AI's easier.

How can I help my child revise logs and exponents for IB Maths?

The single biggest lever is daily short recall practice on the core laws rather than one long weekend session — five minutes of no-calculator fluency drills most days beats a two-hour cram. Topical worksheets that isolate just this subtopic, followed by past-paper questions, work better than generic revision guides.

On RevisionPrep, our Topical Worksheets isolate exponents and logarithms as a standalone set, so your child can drill the exact skill before moving to mixed past-paper questions — useful if a mock has flagged this as a weak topic specifically.

AA vs AI: Logs & Exponents Focus

Aspect	Analysis & Approaches (AA)	Applications & Interpretation (AI)
Emphasis	Algebraic proof & manipulation	Applied modelling & data
Typical context	Solving exponential equations	Growth, decay, finance models
Links to calculus	Yes — differentiating ln x, e^x	Limited, mostly HL
Calculator use	Paper 1 no-calc + Paper 2 calc	Both papers calculator-based

For focused practice, RevisionPrep's Topical Worksheets on Number and Algebra isolate log and exponent questions by difficulty, paired with Mock Papers to test them under exam conditions.