Logs & Exponents (AA SL)
Exponents describe repeated multiplication; a logarithm is the inverse operation, answering the question "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 sits in the exponent - a skill that comes up constantly in growth and decay modelling.
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 syllabus points, examined on both papers.
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 equally to integer, negative, and fractional exponents.
e.g. \(3^2 \times 3^4 = 3^{2+4} = 3^6 = 729\).
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 almost 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.
(a) \(\log_2 8\)
(b) \(\log_{10} 1000\)
(c) \(\ln e^3\)
Worked solution
(a) \(2^3=8\Rightarrow\log_2 8=3.\) A1
(b) \(10^3=1000\Rightarrow\log_{10}1000=3.\) A1
(c) \(\ln e^3=3\) (since \(\ln e=1\)). A1
\(2\log x+\log 3-\log y.\)
(a) Write it as a single logarithm.
(b) Solve \(\log_2 x+\log_2(x-2)=3\).
Worked solution
(a) \(\log\dfrac{3x^2}{y}.\) M1 A1
(b) \(\log_2[x(x-2)]=3\Rightarrow x^2-2x=8\Rightarrow x^2-2x-8=0.\) M1
\((x-4)(x+2)=0,\ x=4\) (reject \(x\) A1
\(=-2\)). 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 to reject invalid roots. After solving a log equation, always check each solution makes every logarithm's argument positive - a root like \(x=-2\) in \(\log_2(x-2)\) must be discarded.
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.
For very large or very small numbers - avoids typing long strings of zeros and prevents rounding errors, which matters when an exponential model gives a huge result.
- Scientific notation means \(a\times10^n\), e.g. \(3.2\times10^8\) or \(4.5\times10^{-3}\).
- Use 2nd → , (EE) to enter the ×10 part: type 3.2 2nd , 8 to enter 3.2×10⁸. Do NOT type ×10^ separately.TI-84
- Use the EE key (or type ×10^ from the keyboard template) to enter scientific notation. Or just type 3.2×10^8 using the ^ key.Nspire
- Use the ×10ᴉ key (EXP key) - type 3.2 then EXP then 8. Do NOT type ×10^ manually.Casio
- To display answers in scientific notation: on TI-84 press MODE and choose SCI; on Casio set the display mode in SET UP.
Tip: A common mistake is typing ×10^ instead of using the EE/EXP key - this gives ×10×... (multiplication, then a power) rather than proper scientific notation.
Faster and safer than algebra for messy exponential equations - and it finds every solution, not just one.
- Graph \(f(x)\) first so you can see how many solutions exist and roughly where they are.
- 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 one root, press ALPHA + ENTER. Move the guess to near a different root and repeat for each solution.TI-84
- Type nSolve(f(x)=0, x, guess) - include a guess or interval e.g. nSolve(f(x)=0, x, 2) or nSolve(f(x)=0, x, {1,5}) to target a specific root.Nspire
- Run-Matrix → SolveN(f(x), x) returns all real roots at once; or use the Equation app for a visual approach.Casio
- For transcendental equations (e.g. \(e^x = 3x\)), graph both sides, count crossings, then use the intersection tool for each one.
- Always verify each solution by substituting back into the original equation.
Tip: The solver finds ONE root near your starting guess - change the guess to find others. The graph shows you how many to expect.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Logs & exponents 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 do I need the change of base formula?
Whenever you need to evaluate a logarithm in a base your calculator doesn't have a dedicated 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.
How do I solve an exponential equation like \(3^x = 20\)?
Take logarithms of both sides. \(3^x = 20\) becomes \(x\ln 3 = \ln 20\), so \(x = \dfrac{\ln 20}{\ln 3}\). This works for any base, and any calculator can evaluate the natural logs at the end.
Can I use my GDC for this topic?
Yes, on Paper 2 - your calculator can evaluate a logarithm in any base and solve exponential equations numerically. On Paper 1 you'll need the laws above without a calculator. See the GDC guide for model-specific instructions.
Sub-topics
Logs & Exponents broken down into its individual skills, each with its own focused page.
Related topics
More Number & Algebra topics from the same AA SL syllabus unit, in case you want to keep going.