Exponents & Logarithms (AA HL)
Exponents (indices) describe repeated multiplication; logarithms are their inverse, answering "what power do I need?" This topic covers the laws for combining exponents and logarithms, converting between the two forms, changing a logarithm's base so it can be evaluated on a calculator, and solving equations where the unknown is an exponent.
What the syllabus says
This topic maps onto two points in the official IB Analysis & Approaches syllabus.
| Code | Syllabus content |
|---|---|
| SL1.5 | Laws of exponents with integer exponents. Introduction to logarithms with base 10 and \(e\). Awareness that \(a^x=b\) is equivalent to \(\log_a b = x\), where \(a>0\), \(b>0\). \(\log_e x = \ln x\). |
| SL1.7 | Laws of exponents with rational exponents. Laws of logarithms: \(\log_a(xy)=\log_a x+\log_a y\), \(\log_a(\tfrac{x}{y})=\log_a x-\log_a y\), \(\log_a(x^m)=m\log_a x\). Change of base of a logarithm. Solving exponential equations, including using logarithms. |
These are core AA SL syllabus points that are also examinable at AA HL.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is an exponent?
An exponent, or index, tells you how many times a number (the base) is multiplied by itself. In \(2^5\), the base is 2 and the exponent is 5. Exponents can be positive, negative, or fractional, and each type follows its own rule for evaluating or simplifying the expression.
e.g. \(2^5 = 2\times2\times2\times2\times2 = 32\).
What is a logarithm?
A logarithm is the inverse of an exponent - it answers "what power do I need?" \(\log_a b = x\) means exactly the same thing as \(a^x = b\). Logarithms let you solve for an unknown exponent, which is otherwise hard to isolate algebraically.
e.g. \(\log_2 8 = 3\), since \(2^3 = 8\).
What are the laws of exponents?
The laws of exponents are rules for combining powers of the same base: multiplying adds the exponents, dividing subtracts them, and raising a power to a power multiplies them. They apply to integer, negative, and fractional exponents alike.
e.g. \(2^3 \times 2^4 = 2^{3+4} = 2^7 = 128\).
What are the laws of logarithms?
The laws of logarithms mirror the laws of exponents: the log of a product is a sum of logs, the log of a quotient is a difference of logs, and the log of a power brings the exponent out the front as a multiplier.
e.g. \(\log_2(8\times4) = \log_2 8 + \log_2 4 = 3+2 = 5\).
What is change of base?
Change of base lets you rewrite a logarithm in any base as a ratio of logarithms in a base your calculator supports, usually base 10 or \(e\). It's essential for evaluating logs like \(\log_5 12\), which most calculators can't compute directly.
e.g. \(\log_5 12 = \dfrac{\ln 12}{\ln 5} \approx 1.544\).
Key formulas
Six formulas cover every question on this topic. The two tables below summarise all of them at a glance - the explanations underneath go into more depth on each one.
Formula reference
The four logarithm laws are on the official formula booklet; the exponent rules are assumed prior knowledge and aren't listed separately.
| Formula | Used for | Booklet? |
|---|---|---|
| \(\log_a(xy) = \log_a x + \log_a y\) | Product law | ✓ Yes |
| \(\log_a\!\left(\tfrac{x}{y}\right) = \log_a x - \log_a y\) | Quotient law | ✓ Yes |
| \(\log_a(x^m) = m\log_a x\) | Power law | ✓ Yes |
| \(\log_a x = \dfrac{\log_c x}{\log_c a}\) | Change of base | ✓ Yes |
| \(a^m \cdot a^n = a^{m+n}\) | Exponent product rule | Not in booklet |
| \((a^m)^n = a^{mn}\) | Exponent power rule | Not in booklet |
Exponents vs logarithms
Every exponent law has a matching logarithm law, because a logarithm is just an exponent written the other way round. This table lines the two up side by side.
| Feature | Exponent form | Logarithm form |
|---|---|---|
| Relationship | \(a^x = b\) | \(\log_a b = x\) |
| Combining (product) | \(a^m \cdot a^n = a^{m+n}\) | \(\log_a(xy) = \log_a x + \log_a y\) |
| Combining (quotient) | \(a^m \div a^n = a^{m-n}\) | \(\log_a\!\left(\tfrac{x}{y}\right) = \log_a x - \log_a y\) |
| Powers | \((a^m)^n = a^{mn}\) | \(\log_a(x^m) = m\log_a x\) |
| Example | \(2^3 = 8\) | \(\log_2 8 = 3\) |
Laws of exponents
These rules only combine powers that share the same base - you can't simplify \(2^3 \times 3^4\) this way, since the bases differ.
Product rule
\[a^m \cdot a^n = a^{m+n}\]
Add the exponents when multiplying powers of the same base.
Not in the formula booklet - prior knowledgeQuotient rule
\[a^m \div a^n = a^{m-n}\]
Subtract the exponents when dividing powers of the same base.
Not in the formula booklet - prior knowledgePower of a power
\[(a^m)^n = a^{mn}\]
Multiply the exponents when raising a power to another power.
Not in the formula booklet - prior knowledgeLaws of logarithms
The laws of logarithms only work when every logarithm in the expression shares the same base \(a\).
Product law
\[\log_a(xy) = \log_a x + \log_a y\]
The log of a product is the sum of the logs.
✓ In the formula bookletQuotient law
\[\log_a\!\left(\tfrac{x}{y}\right) = \log_a x - \log_a y\]
The log of a quotient is the difference of the logs.
✓ In the formula bookletPower law
\[\log_a(x^m) = m\log_a x\]
An exponent inside a log can be brought out the front.
✓ In the formula bookletWorked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
Evaluate exactly.
(a) \(8^{-2/3}\)
(b) \(\left(\dfrac{25}{4}\right)^{-1/2}\)
Worked solution
(a) \(8^{-2/3}=\dfrac{1}{8^{2/3}}.\) M1
\(=\dfrac14.\) A1
(b) \(\left(\dfrac{25}{4}\right)^{-1/2}=\left(\dfrac{4}{25}\right)^{1/2}.\) M1
\(=\dfrac25.\) A1
Given \(\log_2 3 = a\) and \(\log_2 5 = b\), express in terms of \(a\) and \(b\):
(a) \(\log_2 45\)
(b) \(\log_2 \tfrac{8}{15}\)
(c) \(\log_2 0.6\)
Worked solution
(a) \(45=3^2\cdot5.\) M1
\(\log_2 45=2a+b.\) A1
(b) \(\tfrac{8}{15}=\dfrac{2^3}{3\cdot5}.\) M1
\(\log_2\tfrac{8}{15}=3-a-b.\) A1
(c) \(0.6=\tfrac35,\) so \(\log_2 0.6=a-b.\) A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Splitting \(\log(x+y)\) into \(\log x + \log y\). Logarithms don't distribute over addition or subtraction - the product/quotient laws only apply to multiplication and division inside the log.
- Confusing \((a^m)^n\) with \(a^{(m^n)}\). A power of a power multiplies the exponents - \((2^3)^2 = 2^6 = 64\) - which is not the same as a tower of exponents.
- Sign errors with negative indices. \(a^{-n} = \dfrac{1}{a^n}\), not \(-a^n\) - a negative exponent means "reciprocal", not "negative value".
- Forgetting the domain of a logarithm. \(\log_a x\) is only defined for \(x>0\) (with \(a>0\), \(a\neq1\)) - an answer like \(\log_2(-4)\) has no real value and signals an error earlier in the working.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
Work out log to any base directly - useful for solving exponential equations and for logarithmic scales.
- Decide the base \(b\) and the value \(x\) you want \(\log_b(x)\) for.
- Press MATH → logBASE( and enter logBASE(x, b). (Older OS: use log(x)/log(b).)TI-84
- Type log(x, b) directly - the base goes after the comma.Nspire
- Use the log_□□ template (math templates) or type log(b, x) via OPTN → CALC.Casio
Tip: Change of base always works on any calculator: \(\log_b x = \ln x/\ln b = \log x/\log b\).
Faster and safer than algebra for messy exponential equations - and it finds every solution, not just one.
- Rearrange so everything is on one side: \(f(x) = 0\) - or graph both sides as separate functions and find intersections.
- MATH → Solver: enter the expression, type a starting guess close to a root, press ALPHA + ENTER.TI-84
- Type nSolve(f(x)=0, x, guess) - include a guess or interval, e.g. nSolve(f(x)=0, x, {1,5}).Nspire
- Run-Matrix → SolveN(f(x), x) returns all real roots at once.Casio
- Always verify each solution by substituting back into the original equation.
Tip: Graph \(f(x)\) first so you can see how many solutions exist and roughly where they are.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Exponents & logarithms questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What's the difference between an exponent and a logarithm?
An exponent tells you how many times to multiply a base by itself (\(2^3 = 8\)). A logarithm is the inverse - it tells you what exponent you'd need (\(\log_2 8 = 3\)). They describe the same relationship from opposite directions.
When can I use the change of base formula?
Any time you need to evaluate a logarithm in a base your calculator doesn't have a direct button for. \(\log_a x = \dfrac{\log_c x}{\log_c a}\) lets you switch to base 10 or base \(e\), which every calculator supports.
Are log laws examined at HL?
Yes. The laws of logarithms (product, quotient, power) and change of base are core AA syllabus content, examined on both Paper 1 and Paper 2.
Can I use my GDC for this topic?
Yes, on Paper 2 - your calculator can evaluate a logarithm in any base directly. On Paper 1 you'll need the laws above without a calculator. See the GDC guide for model-specific instructions.
Sub-topics
Exponents & Logarithms broken down into its individual skills, each with its own focused page.
Related topics
More Number & Algebra topics from the same AA HL syllabus unit, in case you want to keep going.