Laws of Logarithms and Indices (AI SL)
Before you can solve exponential or log equations, you need to be fluent in the rules for combining powers - multiplying, dividing, raising a power to a power, and handling negative or fractional exponents. This page sets out the index laws you'll use constantly across the course, with worked examples and the mistakes that cost marks. It's part of the broader Exponents & Logarithms topic.
53 questions on this sub-topic.
The key laws
Covered under IB syllabus reference SL1.5: laws of exponents with integer exponents (e.g. \(5^3 \times 5^{-6} = 5^{-3}\), \(6^4 \div 6^3 = 6\), \((2^3)^4 = 2^{12}\)) and an introduction to logarithms, including evaluating them directly.
Laws of indices
\(a^m \times a^n = a^{m+n}\), \(\ \dfrac{a^m}{a^n} = a^{m-n}\), \(\ (a^m)^n = a^{mn}\), and \(a^{-n} = \dfrac{1}{a^n}\).
These are basic algebra you're expected to know, not a formula-booklet entry - they're the toolkit for every simplification question.
Not in the formula booklet - basic algebraCombining or splitting logarithms with the product/quotient/power laws is AHL-only content - see the Solving exponential/log equations page if you're taking AHL.
Need the full syllabus wording and formula-booklet reference table? See Exponents & Logarithms.
Worked examples
Evaluate to 3 significant figures:
(a) \(\ln 7\)
(b) \(e^{2}\)
Worked solution
(a) \(\ln 7 \approx 1.95\) A1
(b) \(e^2 \approx 7.39.\) A1
Simplify \(\dfrac{6x^4 y^3}{2x y^5}\), giving your answer with positive indices.
Worked solution
\(\dfrac{6}{2} = 3.\) A1
\(x^{4-1} = x^3\) and \(y^{3-5} = y^{-2}.\) M1
Move \(y^{-2}\) to the denominator: \(\frac{6x^4y^3}{2xy^5} = \frac{3x^3}{y^2}.\)A1
\(\dfrac{(2x^{3})^{2}}{4x}.\)
(a) Simplify it.
(b) Evaluate the simplified expression when \(x=2.\)
(c) Simplify \(\dfrac{(3x^{2})^{3}}{9x}.\)
Worked solution
(a) \(\dfrac{4x^{6}}{4x}\) M1
\(=x^{5}.\) A1
(b) \(2^{5}\) M1
\(=32.\)A1
(c) \(\dfrac{27x^{6}}{9x}\) M1
\(=3x^{5}.\)A1
Common mistakes
- Sign errors with negative indices. \(a^{-n} = \dfrac{1}{a^n}\), not \(-a^n\) - a negative exponent means "reciprocal", not "negative value".
- Adding exponents on different bases. The rule \(a^m \times a^n = a^{m+n}\) only applies when the bases match. \(x^2 \times y^3\) cannot be combined into a single power - it stays as \(x^2y^3\).
- Applying \((a^m)^n = a^{mn}\) to a sum instead of a product. \((x^2+1)^3\) is not \(x^6+1\) - the power-of-a-power rule only multiplies exponents when the base itself is a single power, not a bracketed sum.
Ready to practise properly?
55 index-and-log-law questions, marked instantly like the real exam. See using your GDC for calculator tips.
Quick answers
What are the three main laws of indices?
\(a^m \times a^n = a^{m+n}\) when multiplying powers of the same base, \(a^m \div a^n = a^{m-n}\) when dividing, and \((a^m)^n = a^{mn}\) when raising a power to another power.
How do you simplify an expression with negative or fractional indices?
A negative index means reciprocal: \(a^{-n} = \tfrac{1}{a^n}\). Apply the index laws first to combine terms, then rewrite any negative exponent in the final answer as a positive-index fraction, since answers are normally expected with positive indices.