Skip to content
Logarithms and Exponentials: Modelling Growth and Decay Problems
🧮 Logarithms and Exponentials: Modelling Growth and Decay Problems
⭐Right—growth and decay. This is the topic where half the class feels fine until the exam switches from a clean curve to something like “a population grows at a rate proportional to its size…” and everyone freezes. But honestly, once you get the rhythm of exponential behaviour, it’s one of the easier modelling tools. It’s predictable, it’s repeatable, and if you’ve practised enough A Level Maths examples and solutions, a lot of the exam questions turn into the same pattern with different labels.
Let’s walk through this slowly, the way I’d explain it at the board, doing that thing where I go, “hang on—let me just check that exponent,” because students always tell me that’s when things suddenly make sense.
🔙 Previous topic:
Previously we covered Modulus Functions: Equations, Inequalities & Graph Transformations, which feeds neatly into the modelling ideas here.
🔻 Exam Context
Exponential models appear everywhere:
- population growth
- radioactive decay
- chain reactions
- compound interest
- cooling and heating
- drug concentration
- bacterial cultures
Examiners love them because they test whether you understand:
- how to translate wording into equations
- when to use exponentials instead of linear models
- how to solve for parameters using logs
- how to justify assumptions (this is where marks sneakily disappear)
And honestly, students lose marks not because the maths is hard, but because they forget to state assumptions like:
- “rate proportional to amount”
- “constant percentage change per unit time”
- “continuous growth model applies”
If you want your modelling to look calm and examiner-friendly, these phrases matter.
🧩 Problem Setup
Let’s anchor everything around one clean model:
P = P_0 e^{kt}
The classic continuous growth/decay equation.
We’ll play with it in both directions—solving forward, extracting constants, and using logs to make sense of parameters.
🧠 Key Ideas Explained
🎯 Step 1 — “Rate proportional to amount” → why exponentials appear everywhere
Let me pause here—students often treat this phrase as just… noise. But it’s the sentence that creates exponentials.
If the change in amount is proportional to the amount:
\frac{dP}{dt} = kP
Solve (you don’t need to show the integration):
P = P_0 e^{kt}
That’s it. The moment the rate depends on the current size, you get exponential behaviour.
- If k > 0, you grow.
- If k < 0, you decay.
This is why bacteria multiply exponentially: more bacteria → faster reproduction → more bacteria → and so on.
🛠️ Step 2 — Using logs to find k (the exam favourite)
Let’s say a population doubles in 8 hours.
Start from:
P = P_0 e^{kt}
At doubling:
2P_0 = P_0 e^{8k}
Cancel:
2 = e^{8k}
Now take natural logs:
\ln 2 = 8k
So:
k = \frac{\ln 2}{8}
This is classic exam working: clean, readable, and impossible to lose method marks on.
A lot of scripts fail here because students jump straight from “doubling every 8 hours” to something vague like k = 0.086 with no justification. Always show the log step.
🗺️ Step 3 — Solving backwards (find time needed)
Let’s flip the logic.
A substance decays according to:
A = 120 e^{-0.03t}
When will it reach 30 units?
Set up:
30 = 120 e^{-0.03t}
Divide both sides:
\frac14 = e^{-0.03t}
Log:
\ln\left(\frac14\right) = -0.03t
So:
t = \frac{\ln(1/4)}{-0.03}
A common student mistake: forgetting that \ln(1/4) is negative. The two negatives cancel, so t is positive. This is one of those A Level Maths revision mistakes to avoid—sign errors in logs will sink a whole modelling question.
🧭 Step 4 — Compound interest vs continuous growth (students mix these up constantly)
Right, quick contrast:
Compound interest formula:
A = P\left(1 + \frac{r}{n}\right)^{nt}
Continuous growth formula:
A = Pe^{rt}
They are not interchangeable unless the context says so.
If a question says “compounded monthly”, use the first.
If it says “rate proportional to amount” or “continuous”, use the exponential.
If the question doesn’t specify?
Examiners expect exponential unless there’s clear discrete behaviour.
A lot of borderline grades lose marks here.
🪫 Step 5 — Half-life problems (decay but with a nicer number)
A radioactive substance halves every 12 days.
Model:
A = A_0 e^{kt}
At half life:
\frac12 = e^{12k}
Log:
\ln\frac12 = 12k
So:
k = \frac{\ln(1/2)}{12}
Students sometimes rewrite \ln(1/2) as -\ln 2, which is fine. But don’t skip steps—in modelling questions, working is everything.
⚡ Step 6 — Combining growth and decay (exam favourite)
Something grows but simultaneously decays (e.g., medication absorption vs breakdown).
The net effect is still exponential, but with a combined constant.
Example:
A chemical increases at 4% per hour but decays at 1.5% per hour.
Net change:
k = 0.04 – 0.015 = 0.025
Model:
A = A_0 e^{0.025t}
Simple. But very easy to misread if the question’s wording is cluttered.
💭 Worked Example — Full realistic modelling walkthrough
Let’s do the kind of messy example examiners adore.
A culture of bacteria grows according to:
N = 180 e^{0.12t}
t in hours.
(a) How many bacteria after 5 hours?
Just substitute:
N = 180 e^{0.6}
A quick mental check:
e^{0.6} is a bit over 1.8, so your answer should be around 320ish.
If you get 9000, you’ve mis-keyed something.
(b) Find how long it takes to reach 1000.
Set:
1000 = 180 e^{0.12t}
Divide:
\frac{1000}{180} = e^{0.12t}
Log:
\ln\left(\frac{1000}{180}\right) = 0.12t
Solve:
t = \frac{\ln(1000/180)}{0.12}
Always write the fraction before taking logs. Examiners want clarity.
(c) State one assumption of this model.
This is the part students forget.
Valid assumptions include:
- growth rate stays proportional
- environmental conditions constant
- no limiting factors
- unlimited resources
- no competition or overcrowding
Write one clearly. You don’t need a paragraph.
❗ Common Errors & Exam Traps
Here are the big ones:
- jumping straight from numbers to k without showing logs
- mixing up growth and decay signs
- treating discrete compound interest as continuous (or vice versa)
- forgetting that \ln(1/4) is negative
- ignoring units (hours vs days)
- forgetting to justify model assumptions
- writing answers with unrealistic rounding
- solving exponentials but forgetting to reverse the log step correctly
A subtle one:
Students forget that k is the continuous rate, not the percentage rate unless the question states otherwise.
🌍 Real-World Link
Exponential models appear in nearly every scientific field: medicine, climate models, computing performance, nuclear physics, finance… basically anything that grows or shrinks in a proportional way. Once you can solve P = P_0 e^{kt} calmly, huge parts of applied maths become accessible.
🚀 Next Steps
If exponential modelling still feels a bit slippery—or if logs keep tripping you up when rearranging—the A Level Maths Revision Course with guided practice walks through dozens of real modelling examples step by step, with the reasoning spelled out the way examiners like.
📏 Recap Table
- \frac{dP}{dt} = kP → gives exponential
- P = P_0 e^{kt}
- Use logs to extract k
- “<” half-life → negative k
- Combine rates when growth + decay
- Use discrete or continuous depending on context
- Always justify assumptions
👤Author Bio – S Mahandru
I’ve taught exponential modelling for years, and honestly, once you slow down and treat logs as a tool rather than a threat, this topic becomes one of the most predictable parts of the course.
🧭 Next topic:
Once you’ve used logarithms and exponentials to model continuous growth and decay, the natural next step is sequences and series, where those same ideas get applied discretely — adding change step by step using arithmetic and geometric patterns and expressing it cleanly with sigma notation.
❓ FAQ Section
When do I actually use logs in these modelling problems?
Pretty much anytime the unknown is sitting awkwardly in the exponent, that’s your cue to reach for logs. Students often try to “rearrange” exponentials by doing random algebra first, and it never works—logs are the only clean way to bring the power down where you can solve it. The moment you see something like e^{kt} and you’re solving for k or t, log both sides and let the structure do the heavy lifting. And honestly, the more you do it, the more routine it feels—half the panic disappears once you realise it’s always the same move. That’s why examiners include these steps: they want to see you apply the log at the right moment, not as an afterthought.
Why is k negative for decay? It feels like a trick.
It’s not a trick at all—it’s just the maths reflecting the story correctly. If something is shrinking, then every extra unit of time should reduce the amount, not increase it, so the model needs a mechanism to push the curve downward. A negative k achieves exactly that by turning e^{kt} into a fraction that gets smaller as time increases. A lot of students try to “force” decay by sticking a minus sign in the wrong place, but if the situation genuinely involves proportional decrease, k must be negative by definition. Once you accept that the sign is part of the modelling, not decoration, decay problems feel much calmer.
Do I really need to justify assumptions in modelling questions?
Yes—especially in the longer 6–8 mark questions where the examiner is checking whether you understand why the model works, not just how to crunch the numbers. A single clear sentence like “the rate is proportional to the amount” or “environmental conditions remain constant” can carry actual method marks. Students lose marks every year because they assume assumptions don’t matter, which sounds ridiculous but happens in exam pressure. The whole point of a model is that it’s only valid because certain conditions stay stable; the examiner just wants you to show that you know this. Think of it as proving you didn’t solve the question on autopilot.