Laws of Logarithms and Indices (AA HL)
Before you can solve exponential or log equations confidently, you need the underlying rules cold: how powers combine, how negative and fractional indices behave, and how the product, quotient and power laws let you split or merge logarithms. This page sets out those rules with two worked examples and the slips that trip up even careful students. It's part of the broader Exponents & Logarithms topic.
19 questions on this sub-topic.
The key laws
Covered under IB syllabus references SL1.5 (laws of exponents with integer exponents, and logarithms as the inverse of exponentials, with \(\log_e x = \ln x\)) and SL1.7 (laws of exponents with rational exponents, and the laws of logarithms). Neither set of laws is printed in the formula booklet - both need to be memorised.
Laws of indices
\(a^m a^n = a^{m+n}, \quad (a^m)^n = a^{mn}, \quad a^{-n} = \dfrac{1}{a^n}, \quad a^{1/n} = \sqrt[n]{a}\)
Apply these to expand or simplify any expression with powers before touching the numbers.
Laws of logarithms
\(\log_a(xy) = \log_a x + \log_a y, \quad \log_a\!\left(\dfrac{x}{y}\right) = \log_a x - \log_a y, \quad \log_a(x^m) = m\log_a x\)
These only combine logs with the same base - use the change-of-base formula first if the bases don't match.
Need the full syllabus wording and formula-booklet reference table? See Exponents & Logarithms. Once you're confident with the laws, move on to Exponential and Log Equations.
Worked examples
Simplify, giving each answer with positive indices.
(a) \(\dfrac{(2x^2 y^{-1})^3}{4x y}\)
(b) \(\left(27 a^6\right)^{2/3}\)
Worked solution
(a) \(\dfrac{(2x^2 y^{-1})^3}{4x y} = \dfrac{8x^6 y^{-3}}{4xy}\) M1
\(= 2x^{5} y^{-4} = \dfrac{2x^5}{y^4}.\) A1
(b) \(\left(27 a^6\right)^{2/3} = 27^{2/3}(a^6)^{2/3}\) M1
\(= 9a^4.\) A1
Write \(2\log x+\tfrac12\log y-\log z\) as a single logarithm.
Worked solution
\(2\log x + \tfrac12\log y - \log z\) uses \(n\log a=\log a^n.\) M1 This gives \(\log x^2 + \log y^{1/2} - \log z.\) A1
\(\log x^2 + \log y^{1/2} - \log z = \log\!\left(x^2\cdot y^{1/2}\right) - \log z.\) M1
Step 2b - write as a single log. \(= \log\!\left(\dfrac{x^2\sqrt{y}}{z}\right).\) 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".
- Combining logs with different bases. \(\log_2 x + \log_3 y\) cannot be merged with the product law - the product, quotient and power laws only work when every log in the expression shares the same base.
- Forgetting a bracket needs distributing over every factor. \((2x^2y^{-1})^3\) means the power 3 applies to the 2, the \(x^2\), and the \(y^{-1}\) separately - a common slip is to cube only the variable part and leave the coefficient unraised.
Ready to practise properly?
19 index and log law questions, marked instantly like the real exam.
Quick answers
What are the three main laws of logarithms?
The product law \(\log_a(xy) = \log_a x + \log_a y\), the quotient law \(\log_a\!\left(\tfrac{x}{y}\right) = \log_a x - \log_a y\), and the power law \(\log_a(x^m) = m\log_a x\).
How do you simplify an expression with negative or fractional indices?
Use \(a^{-n} = \tfrac{1}{a^n}\) to turn negative indices into positive ones, and \(a^{m/n} = \left(\sqrt[n]{a}\right)^m\) to interpret fractional indices as roots and powers combined.