Exponential & Logarithmic Models (AI HL)
Anything that grows or decays by a fixed percentage - bacteria, radioactive isotopes, a cooling drink, compound interest - is modelled with an exponential function, not a straight line. This topic covers exponential growth and decay, half-life, natural logarithmic models with their own distinct shape, and the technique of taking logs of both variables to turn a curved relationship into a straight line you can fit by regression.
What the syllabus says
This topic maps onto two points in the official IB Applications & Interpretation syllabus - the SL exponential model and the AHL extension to natural logarithmic models.
| Code | Syllabus content |
|---|---|
| SL2.5 | Exponential growth and decay models: \(f(x)=ka^x+c\), \(f(x)=ka^{-x}+c\) (for \(a>0\)), \(f(x)=ke^{rx}+c\). Equation of a horizontal asymptote. |
| AHL2.9 | Exponential models to calculate half-life. Natural logarithmic models: \(f(x)=a+b\ln x\). |
AHL2.9 also covers sinusoidal, logistic and piecewise models, which are treated as their own topics on this site.
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 exponential growth/decay model?
An exponential model has the input variable in the exponent, so the quantity changes by a fixed percentage each unit of \(x\) rather than by a fixed amount. \(f(x)=ka^x+c\) grows if \(a>1\) and decays if \(0
e.g. \(N(t)=50e^{0.2t}\): \(N(5) = 50e^{1} \approx 135.9.\)
What is a natural logarithmic model?
A natural logarithmic model \(f(x)=a+b\ln x\) has the input variable inside a logarithm. It rises (or falls) quickly for small \(x\) and levels off as \(x\) grows, and it's only defined for \(x>0\).
e.g. \(f(x)=2+3\ln x\): \(f(e) = 2+3(1) = 5,\) since \(\ln e = 1.\)
What is half-life?
Half-life is the time it takes a decaying quantity to fall to half its value. For \(A(t)=A_0e^{-kt}\), setting \(A=\tfrac12A_0\) and solving gives \(k=\dfrac{\ln2}{\text{half-life}}\).
e.g. Half-life \(=10\) years: \(k=\dfrac{\ln2}{10}\approx0.0693.\)
What is a horizontal asymptote (exponential model)?
In \(f(x)=ka^x+c\), the term \(ka^x\) shrinks towards zero as \(x\to\pm\infty\) (depending on the sign of the exponent), leaving \(f(x)\to c\). The line \(y=c\) is the horizontal asymptote - not \(y=0\).
e.g. \(f(x)=5e^{-0.3x}+20\): as \(x\to\infty\), \(f(x)\to20\), so the asymptote is \(y=20.\)
What does linearising with logs mean?
Taking logs of a power model \(y=ax^n\) gives \(\log y = \log a + n\log x\), a straight line in \(\log x\) and \(\log y\). The gradient of that line is \(n\); the intercept is \(\log a\).
e.g. Gradient \(2.5\), intercept \(0.6\): \(n=2.5\), \(a=10^{0.6}\approx3.98.\)
Key formulas
Two model forms and two derived relationships cover almost every question on this topic. The tables below summarise all of them at a glance - the explanations underneath go into more depth on each one.
Formula reference
The AI formula booklet lists the exponential and natural logarithmic model forms directly. The half-life relationship and change of base are algebraic consequences you're expected to derive, not look up as a single formula.
| Formula | Used for | Booklet? |
|---|---|---|
| \(f(x)=ka^x+c\) or \(f(x)=ke^{rx}+c\) | Exponential growth/decay model | ✓ Yes |
| \(f(x)=a+b\ln x\) | Natural logarithmic model | ✓ Yes |
| \(k=\dfrac{\ln2}{\text{half-life}}\) | Decay constant from half-life | Not in the formula booklet - derived from the half-life definition |
| \(\log_a x = \dfrac{\ln x}{\ln a}\) | Change of base (for evaluating on a GDC) | ✓ Yes |
Exponential vs logarithmic models
These two model types are inverses of each other, and it shows in every one of their key features.
| Feature | Exponential \(f(x)=ka^x+c\) | Logarithmic \(f(x)=a+b\ln x\) |
|---|---|---|
| Where the variable sits | In the exponent | Inside the logarithm |
| Domain | All real \(x\) | \(x>0\) |
| Asymptote | Horizontal, \(y=c\) | Vertical, \(x=0\) |
| Shape | Speeds up (growth) or flattens (decay) | Rises quickly then flattens out |
Exponential growth and decay
The same general form covers both directions - the sign of the rate constant decides which one you get.
General form
\[f(x)=ke^{rx}+c\]
\(k\) scales the exponential part, \(r\) sets the growth (\(r>0\)) or decay (\(r<0\)) rate, and \(c\) shifts the asymptote.
✓ In the formula bookletHorizontal asymptote
As \(x\to\infty\) (growth) or \(x\to-\infty\) (decay), \(e^{rx}\to0\), so \(f(x)\to c\) - always check the sign of \(c\), not just \(k\).
Half-life
\[k=\frac{\ln2}{\text{half-life}}\]
Derived by setting \(A(t)=\tfrac12A_0\) in \(A_0e^{-kt}\) and solving for \(t\).
Natural logarithmic models
These behave very differently from exponentials, even though the two are algebraically linked.
General form
\[f(x)=a+b\ln x\]
\(a\) shifts the curve vertically, \(b\) controls how steeply it rises or falls.
✓ In the formula bookletDomain restriction
\(\ln x\) is undefined for \(x\le0\), so the model's domain is always \(x>0\) - there's a vertical asymptote at \(x=0\).
Linearising a power model
For \(y=ax^n\), taking logs of both sides gives a straight line \(\log y = n\log x + \log a\) - fit this by linear regression on the logged data.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
A bacterial culture grows as \(N(t) = 200 e^{0.35t}\), \(t\) in hours.
(a) Find the initial population.
(b) Find the population after 6 hours.
(c) Find when the population reaches 5000.
Worked solution
(a) \(N(0) = 200.\) A1
(b) \(N(6) = 200 e^{2.1}\) M1
\(\approx 1633.\) A1
(c) \(e^{0.35t} = 25 \Rightarrow t = \dfrac{\ln 25}{0.35}\) M1
\(\approx 9.20\) h. A1
Carbon-14 decays as \(A(t) = A_0 e^{-kt}\) with half-life 5730 years.
(a) Find \(k\), to 3 s.f.
(b) A sample has \(35\%\) of its original C-14. Estimate its age.
Worked solution
(a) \(0.5 = e^{-k(5730)} \Rightarrow k = \dfrac{\ln 2}{5730}\) M1
\(\approx 1.21\times10^{-4}\) /yr. A1
(b) \(0.35 = e^{-kt}\) M1
\(\Rightarrow t = \dfrac{-\ln 0.35}{k}\) M1
\(\approx 8680\) years. 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.
- Forgetting the domain of a log model. A natural log model \(f(x)=a+b\ln x\) is undefined for \(x\le0\) - always state \(x>0\) when asked for the domain.
- Confusing growth and decay. It's the sign of the rate constant \(r\) (or \(k\)) that determines growth or decay, not the size of the base \(a\) - \(e^{-0.1t}\) is decay even though \(e\) itself is bigger than 1.
- Misreading the asymptote. In \(f(x)=ka^x+c\), the horizontal asymptote is \(y=c\), not \(y=0\) - the constant term shifts the whole curve, including where it levels off.
- Rounding too early. Rounding \(k\) or \(a\) partway through a multi-step problem compounds the error in later steps - carry exact values (e.g. \(\ln2/5730\)) through the calculation and round only the final answer.
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: \(\log_b x = \ln x/\ln b = \log x/\log b\).
The heart of AI modelling - find the best-fitting curve for a data set, not just a straight line.
- Enter the data in two lists (x and y).
- STAT → EDIT to enter L1/L2, then STAT → CALC → QuadReg / CubicReg / ExpReg / PwrReg / SinReg.TI-84
- In a Lists & Spreadsheet page enter the data, then menu → Statistics → Stat Calculations → choose the regression type.Nspire
- Main menu → Statistics, enter the data in lists, then CALC → REG and pick X² / X³ / Exp / Power / Sin.Casio
Tip: Turn DiagnosticOn (TI-84: 2nd → 0 → DiagnosticOn) to see R². Choose the model with the best R² that also makes sense for the context.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Exponential and logarithmic model questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What's the difference between exponential and logarithmic models?
An exponential model \(f(x) = ka^x + c\) has the variable in the exponent and describes growth or decay that speeds up or slows down over time. A logarithmic model \(f(x) = a + b \ln x\) has the variable inside a logarithm and describes a quantity that rises quickly at first, then levels off - it's the inverse relationship.
How do I find the half-life from a decay model?
Set the model equal to half its initial value and solve for \(t\), or use \(k = \ln 2 / (\text{half-life})\) directly if the model is written as \(A_0 e^{-kt}\). Both approaches use the fact that the ratio of quantity remaining is exactly 0.5 at one half-life.
Why does a natural log model have a restricted domain?
'ln x' is only defined for \(x > 0\), so any model of the form \(f(x) = a + b \ln x\) automatically excludes \(x \le 0\) from its domain - there's a vertical asymptote at \(x = 0\) that the graph approaches but never crosses.
Can my GDC fit an exponential or logarithmic model to data directly?
Yes - enter the data into two lists and run ExpReg (or LnReg, where available) to get the model's coefficients and R² value directly, without solving simultaneous equations by hand. See the GDC guide for model-specific instructions.
Sub-topics
Exponential & Logarithmic Models broken down into its individual skills, each with its own focused page.
Related topics
More Functions & Modelling topics from the same AI HL syllabus unit, in case you want to keep going.