Exponentials & Logarithms (AA SL)
Exponentials describe repeated multiplication of a base by itself; logarithms are 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.
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 content, examinable on Paper 1 and Paper 2 at both AA SL and 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 \(3^4\), the base is 3 and the exponent is 4. Exponents can be positive, negative, or fractional, and each type follows its own rule for evaluating or simplifying the expression.
e.g. \(3^4 = 3\times3\times3\times3 = 81\).
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. \(\log_2 32 = 5\), since \(2^5 = 32\).
What is the natural logarithm?
The natural logarithm, written \(\ln x\), is the logarithm with base \(e \approx 2.718\). It's introduced alongside base-10 logarithms and behaves exactly like any other logarithm - it just uses \(e\) as its base.
e.g. \(\ln e^2 = 2\), since \(e^2 = e^2\).
What is an exponential equation?
An exponential equation has the unknown in the exponent, such as \(2^x = 32\). Simple ones can be solved by writing both sides as powers of the same base and equating exponents; harder ones need logarithms to bring the exponent down.
e.g. \(2^x = 32 = 2^5\), so \(x=5\).
What is a horizontal asymptote?
A horizontal asymptote is a horizontal line that a graph approaches but never reaches as \(x\) goes to \(+\infty\) or \(-\infty\). Every exponential graph \(y=a^x\) has a horizontal asymptote, since \(a^x\) gets arbitrarily close to 0 but is never equal to 0.
e.g. \(y=2^x+1\) has horizontal asymptote \(y=1\), since \(2^x\to0\) as \(x\to-\infty\).
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 | \(2^5 = 32\) | \(\log_2 32 = 5\) |
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.
Consider the curve \(y=3^{x}.\)
(a) Find the \(y\)-intercept.
(b) State the horizontal asymptote.
Worked solution
(a) At \(x=0,\ y=1.\) A1
(b) \(y=0.\) A1
\(3\cdot2^{x}=24.\)
(a) Solve for \(x\).
(b) Hence solve \(3\cdot2^{x}=48.\)
Worked solution
(a) \(2^x=8\Rightarrow x=3.\) M1
\(x=3.\) A1
(b) \(2^x=16\Rightarrow x=4.\) 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
On calculator papers, your GDC can evaluate any logarithm directly (most models have a log button that accepts a base, or you can use the change-of-base formula with the LOG or LN key) and can solve exponential equations numerically using an equation solver or by graphing both sides and finding the intersection. On non-calculator papers you'll need the laws above worked by hand instead. See the full GDC guide for model-specific button sequences across TI-84, TI-Nspire, and Casio.
Ready to practise properly?
Exponentials & 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 (\(3^4 = 81\)). A logarithm is the inverse - it tells you what exponent you'd need (\(\log_3 81 = 4\)). 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.
Do I need my GDC for this topic?
Some questions are calculator-free - you're expected to know the laws and evaluate simple powers and logs by hand. Other questions, especially ones asking for a decimal answer, expect you to use your GDC's log button or solver.
Is this content examined at both SL and HL?
Yes. Laws of exponents, laws of logarithms, change of base, and solving exponential equations are core content shared by both Analysis & Approaches SL and HL, examined on Paper 1 and Paper 2.
Sub-topics
Exponentials & Logarithms broken down into its individual skills, each with its own focused page.
Related topics
More Functions topics from the same AA SL syllabus unit, in case you want to keep going.