Exponents & Logarithms (AI SL)
Exponents (indices) describe repeated multiplication, and this topic starts by revising the laws for combining them with whole-number powers. From there it introduces logarithms as the inverse operation - the tool that lets you solve for an unknown exponent, which underpins every population, decay, and growth model later in the course. AI SL keeps this light on algebra: logarithms are evaluated numerically on your GDC rather than derived by hand, so the real skill is setting up the right equation and reading the calculator correctly.
What the syllabus says
This topic maps onto one point in the official IB Applications & Interpretation syllabus.
| Code | Syllabus content |
|---|---|
| SL1.5 | Laws of exponents with integer exponents, e.g. \(5^3 \times 5^{-6} = 5^{-3}\), \(6^4 \div 6^3 = 6\), \((2^3)^4 = 2^{12}\). |
| SL1.5 | Introduction to logarithms with base 10 and \(e\). Awareness that \(a^x = b\) is equivalent to \(\log_a b = x\), where \(a>0,\ b>0\). \(\log_e x = \ln x\). |
| SL1.5 | Numerical evaluation of logarithms using technology. |
SL1.5 sits in the Number & Algebra unit and links directly to exponential growth and decay models later in the Functions unit (SL2.5) - the full laws of logarithms (product, quotient, power, change of base) are only examined at AI 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. AI SL only requires integer exponents - positive, negative, or zero.
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 \(a^x = b\). AI SL introduces this with base 10 and base \(e\).
e.g. \(\log_{10} 1000 = 3\), since \(10^3 = 1000\).
What are the laws of exponents?
The laws of exponents are rules for combining integer powers of the same base: multiplying adds the exponents, dividing subtracts them, and raising a power to a power multiplies them.
e.g. \(2^3 \times 2^4 = 2^{3+4} = 2^7 = 128\).
What is a natural logarithm?
The natural logarithm, written \(\ln x\), is a logarithm with base \(e \approx 2.71828\). It's written \(\log_e x = \ln x\) and appears constantly in growth and decay models because of how \(e\) behaves under differentiation.
e.g. \(\ln(e^3) = 3\), since raising \(e\) to the power 3 and then taking \(\ln\) undoes it.
How do you solve an exponential equation using logs?
Isolate the exponential term on one side, then take logs of both sides so the unknown exponent comes down to become a regular multiplier, leaving an equation you can solve directly.
e.g. \(200(1.15)^t = 1000 \Rightarrow (1.15)^t = 5 \Rightarrow t = \dfrac{\ln 5}{\ln 1.15} \approx 11.5\).
Key formulas
Only a handful of relationships are needed here - the tables below summarise them, and the cards underneath go into more depth on each one.
Formula reference
The exponent-logarithm equivalence is on the official formula booklet; the integer exponent rules are assumed prior knowledge and aren't listed separately.
| Formula | Used for | Booklet? |
|---|---|---|
| \(a^x = b \iff \log_a b = x\) | Definition of a logarithm | ✓ Yes |
| \(\log_e x = \ln x\) | Natural logarithm notation | ✓ Yes |
| \(a^m \cdot a^n = a^{m+n}\) | Exponent product rule | Not in the formula booklet - prior knowledge |
| \(a^m \div a^n = a^{m-n}\) | Exponent quotient rule | Not in the formula booklet - prior knowledge |
| \((a^m)^n = a^{mn}\) | Exponent power rule | Not in the formula booklet - prior knowledge |
Exponential form vs logarithmic form
Every exponential statement has a matching logarithmic statement, because a logarithm is just an exponent written the other way round.
| Feature | Exponential form | Logarithmic form |
|---|---|---|
| Relationship | \(a^x = b\) | \(\log_a b = x\) |
| Base 10 example | \(10^3 = 1000\) | \(\log_{10} 1000 = 3\) |
| Base \(e\) example | \(e^0 = 1\) | \(\ln 1 = 0\) |
| Solving for an unknown exponent | Take logs of both sides | Already isolated |
Laws of exponents (integer 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 knowledgeLogarithms and equations
AI SL treats logarithms as a numerical tool for solving exponential equations, not as an object to manipulate algebraically with a full set of log laws.
Introducing logs
A logarithm answers "what power gives this value?" AI SL uses base 10 (\(\log\)) and base \(e\) (\(\ln\)) specifically.
\(a^x=b \iff \log_a b = x\), for \(a>0,\ b>0\).
✓ In the formula bookletEvaluating on your GDC
The syllabus expects numerical evaluation using technology - type the value straight into the log or ln key rather than working it out by hand.
\(\ln(0.5) \approx -0.693\), read straight off the calculator.
Not in the formula booklet - calculator skillSolving exponential equations
Isolate the exponential term, then take logs of both sides to bring the unknown exponent down as a multiplier.
\(80e^{-0.05t}=40 \Rightarrow t = \dfrac{\ln 0.5}{-0.05} \approx 13.9\).
Not in the formula booklet - techniqueWorked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
A bacteria population is \(N = 200(1.15)^t\) where \(t\) is in hours.
(a) State the initial population.
(b) Find the population after 10 hours.
(c) Find when the population reaches 1000.
Worked solution
(a) Initial population. At \(t=0\), \((1.15)^0 = 1\), so \(N = 200.\) A1
(b) After 10 hours. \(N = 200(1.15)^{10} = 200(4.0456)\) M1
\(\approx 809.\) A1
(c) Reaching 1000. \(200(1.15)^t = 1000 \Rightarrow (1.15)^t = 5 \Rightarrow t = \frac{\ln 5}{\ln 1.15}\) M1
\(\approx 11.5 \text{ hours}.\) A1
A drink cools as \(T = 20 + 60 e^{-0.1t}\) °C, \(t\) in minutes.
(a) Find the initial temperature.
(b) Find the temperature after 15 minutes.
(c) Find when the temperature reaches 40 °C.
Worked solution
(a) Initial temperature. At \(t=0\), \(e^0=1\): \(T = 20 + 60 = 80\) °C. A1
(b) After 15 minutes. \(T = 20 + 60e^{-0.1(15)} = 20 + 60e^{-1.5} = 20 + 60(0.22313)\) M1 \(\approx 33.4 \text{ °C}.\) A1
(c) Reaching 40 °C: \(20 + 60e^{-0.1t} = 40 \Rightarrow e^{-0.1t} = \tfrac{20}{60} = \tfrac13.\) M1
\(t = \dfrac{\ln 3}{0.1} \approx 11.0 \text{ min}.\) A1 The \(+20\) is the room temperature the drink cools towards (the asymptote).
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.
- 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".
- Taking logs before isolating the exponential term. In \(20+60e^{-0.1t}=40\), you must subtract the 20 first to get \(60e^{-0.1t}=20\) before taking logs - taking \(\ln\) of the whole equation as it stands doesn't simplify anything.
- Forgetting the domain of a logarithm. \(\log_a x\) is only defined for \(x>0\) (with \(a>0,\ a\neq1\)) - an answer like \(\ln(-4)\) 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.
For very large or very small numbers - avoids typing long strings of zeros and prevents rounding errors, essential for log-scale contexts like sound intensity (\(10^{-12}\) W/m\(^2\)) or earthquake amplitude.
- 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 \(3.2\times10^8\). 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 ×10ˣ 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 essential in AI, where the syllabus expects logarithms to be evaluated numerically rather than manipulated by hand.
- 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
- 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?
Exponents & logarithms questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What's the relationship between an exponent and a logarithm?
A logarithm is the inverse of an exponent - it answers the question "what power do I need?" \(a^x = b\) means exactly the same thing as \(\log_a b = x\). AI SL introduces this idea with base 10 and base \(e\) (written \(\ln x\)).
How do I evaluate a logarithm like log base 5 of 12 on my GDC?
Every GDC evaluates \(\log_{10}\) and \(\ln\) directly; some also have a log-base template for any base. AI SL only requires numerical evaluation using technology, so you can type the expression straight into your calculator rather than deriving it by hand.
How do I solve an exponential equation like 200(1.15)^t = 1000?
Isolate the exponential term first: \((1.15)^t = 5\). Then take logs of both sides and use the power law of logs to bring the exponent down: \(t = \dfrac{\ln 5}{\ln 1.15} \approx 11.5\). You can also solve it directly with your GDC's numerical solver.
Can I use my GDC for this topic?
Yes - every question on this topic is calculator-permitted, and the syllabus explicitly expects numerical evaluation of logarithms using technology rather than by hand. Your GDC's ln and log keys, plus its numerical equation solver, cover almost everything here. 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 AI SL syllabus unit, in case you want to keep going.