Edexcel IAL P3 Exponentials and Logarithms: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

P3 exponentials and logarithms is about $e^{x}$ and $\ln x$ , which undo each other: $\ln (e^{x}) = x$ and $e^{\ln x} = x$ for $x > 0$ . You solve equations with them, use exponential growth and decay models, and turn $y = a x^{n}$ or $y = k b^{x}$ into straight-line graphs by taking logs.

P3 · Pure Mathematics 3
Calculator allowed
3 worked examples
30 practice questions
Tested in 4 of the last 4 papers

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Notes
Formulas
Worked examples
Mistakes
FAQ

P3 Exponentials and Logarithms notes

eˣ and ln x

$y = e^{x}$ passes through $(0, 1)$ with asymptote $y = 0$ ; $y = \ln x$ passes through $(1, 0)$ with asymptote $x = 0$ . Each is the reflection of the other in $y = x$ . To solve $e^{2 x - 1} = 5$ , take ln of both sides: $2 x - 1 = \ln 5$ , so $x = \frac{1 + \ln 5}{2} \approx 1.30$ .

Equations that are quadratics in eˣ

$e^{2 x} = (e^{x})^{2}$ , so $e^{2 x} - 7 e^{x} + 12 = 0$ is a quadratic in $u = e^{x}$ : $u = 3$ or $u = 4$ , so $x = \ln 3$ or $x = \ln 4$ . Reject any negative value of $u$ , because $e^{x} > 0$ for every $x$ .

Growth and decay models

In $N = N_{0} e^{k t}$ , $N_{0}$ is the starting value and $k > 0$ means growth, $k < 0$ decay. For $N = 200 e^{0.3 t}$ , the time for $N$ to reach 1000 is $t = \frac{\ln 5}{0.3} \approx 5.36$ . In $θ = 20 + 60 e^{- 0.1 t}$ the starting value is 80 and the long-term value is 20, because $e^{- 0.1 t} \to 0$ .

Straight-line graphs from logs

Taking logs turns two curves into lines. $y = a x^{n}$ gives $\log y = n \log x + \log a$ : plot $\log y$ against $\log x$ , gradient $n$ , intercept $\log a$ . $y = k b^{x}$ gives $\log y = x \log b + \log k$ : plot $\log y$ against $x$ , gradient $\log b$ , intercept $\log k$ . If the line of $\log_{10} y$ against $\log_{10} x$ passes through $(0, 0.3)$ and $(2, 1.5)$ , then $n = 0.6$ and $a = 10^{0.3} \approx 2.00$ .

Key formulas

Exponential form In the formula booklet: $e^{x \ln a} = a^{x}$
Log laws Learn it: $\log_{a} x y = \log_{a} x + \log_{a} y, \log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y, \log_{a} x^{k} = k \log_{a} x$
Change of base In the formula booklet: $\log_{a} x = \frac{\log_{b} x}{\log_{b} a}$

In the formula booklet = printed in Pearson's IAL formulae booklet, so know where to find it. Learn it = not printed there, so learn it by heart. All P3 formulas on one page (free PDF) →

P3 Exponentials and Logarithms worked examples

Each mark scheme is checked line by line before it is published here. M marks are for method, A marks for accurate answers that follow from them, B marks for independent results.

Worked example 1 · 5 marks · hard

The variables $x$ and $y$ satisfy the equation $y = a b^{x}$ . A graph of $\ln y$ against $x$ is plotted and forms a straight line passing through $(0, 2)$ and $(4, 10)$ . Find the exact values of the constants $a$ and $b$ .

Mark scheme

M1 for taking natural logarithms of both sides to get $\ln y = \ln a + x \ln b$ .
M1 for calculating the gradient of the line: $\ln b = \frac{10 - 2}{4 - 0} = 2$ .
A1 for solving to find exactly $b = e^{2}$ .
M1 for identifying the y-intercept from the coordinate $(0, 2)$ , so $\ln a = 2$ .
A1 for solving to find exactly $a = e^{2}$ .

5 steps · 1 to fill in

Worked example 2 · 4 marks · hard

Solve the equation $2 \ln x = \ln (3 x + 4)$ .

Mark scheme

M1 for applying the power law to the LHS: $\ln (x^{2}) = \ln (3 x + 4)$ .
M1 for equating the arguments: $x^{2} = 3 x + 4$ .
M1 for rearranging to $x^{2} - 3 x - 4 = 0$ and finding roots $x = 4$ and $x = - 1$ .
A1 for rejecting $x = - 1$ (since $\ln x$ must have $x > 0$ ) and giving $x = 4$ .

4 steps · 1 to fill in

Worked example 3 · 4 marks · medium

Solve the equation $e^{2 x} + 4 e^{x} - 12 = 0$ , giving your answer(s) in exact form.

Mark scheme

M1 for identifying the hidden quadratic and factorising: $(e^{x} + 6) (e^{x} - 2) = 0$ .
M1 for extracting the roots $e^{x} = 2$ and $e^{x} = - 6$ .
M1 for rejecting $e^{x} = - 6$ since an exponential function must be positive.
A1 for the exact valid solution $x = \ln 2$ .

4 steps

Play with it Exponential growth model: drag the sliders

The table shows the number of bacteria (in thousands) after x hours. Change a and b so that y = a·bˣ fits the data (the check wants R² of at least 0.99). What do a and b mean in this context?

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Common P3 exponentials and logarithms mistakes

Taking ln term by term. $\ln (e^{x} + 3)$ is not $x + \ln 3$ . Rearrange to a single $e^{\dots}$ on one side before taking logs.
Keeping a negative eˣ. $e^{x} = - 2$ has no solution: say so and reject it.
Reading the intercept as a. In $\log y = n \log x + \log a$ the intercept is $\log a$ , so $a$ is $10$ (or $e$ ) to the power of the intercept.
The long-term value. As $t \to \infty$ , $e^{- k t} \to 0$ for $k > 0$ : the model tends to its constant term, not to zero.

Every P3 mistake from the examiner reports, with the paper it came from →

On your calculator: how to solve equations, polynomials and simultaneous equations. A calculator is allowed in every IAL unit.

P3 Exponentials and Logarithms FAQ

Is ln x the same as log base e?

Yes. ln x means logₑ x, where e is about 2.718. Every law of logarithms works for ln in the same way.

How do I find a and n from a log graph?

For y = axⁿ, plot log y against log x. The gradient of the straight line is n and the intercept is log a, so a is 10 to the power of the intercept (or e to the power of it if you used ln).

Is a calculator allowed in IAL P3?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P3 included. Method marks are for the working you write down, so show it.

How long is the IAL P3 exam?

Each IAL maths unit, P3 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does exponentials and logarithms come up in P3?

In our mapping of the 4 Pearson P3 papers from June 2024 to June 2025, exponentials and logarithms was tested in 4 of them (9 questions in all). Any topic can be examined in any series.

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More P3: Functions · Numerical Methods · Further Differentiation · Further Integration · Algebraic Methods · Trigonometric Functions · Trigonometric Addition Formulae

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Edexcel IAL P3 Exponentials and Logarithms: exam questions and answers, notes and worked examples

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P3 Exponentials and Logarithms notes

eˣ and ln x

Equations that are quadratics in eˣ

Growth and decay models

Straight-line graphs from logs

Key formulas

P3 Exponentials and Logarithms worked examples

Worked example 1 · 5 marks · hard

Mark scheme

Worked example 2 · 4 marks · hard

Mark scheme

Worked example 3 · 4 marks · medium

Mark scheme

Common P3 exponentials and logarithms mistakes

P3 Exponentials and Logarithms FAQ

Is ln x the same as log base e?

How do I find a and n from a log graph?

Is a calculator allowed in IAL P3?

How long is the IAL P3 exam?

How often does exponentials and logarithms come up in P3?

Next steps

Your P3 revision kit