Solve the equation $\log_2 x + \log_x 2 = 2.5$ for $x > 0, x \ne 1$.

M1 for applying the reciprocal rule: $\log_x 2 = \frac{1}{\log_2 x}$. M1 for making the substitution $u = \log_2 x$ to form $u + \frac{1}{u} = 2.5$. M1 for multiplying by $2u$ to form the quadratic $2u^2 - 5u + 2 = 0$ and solving it. M1 for finding the roots $u = 2$ and $u = \frac{1}{2}$. A1 for converting back to $x$ and finding both solutions: $x = 4$ and $x = \sqrt{2}$.

A Level Math Revision

Edexcel IAL P2 Logarithms and Exponentials: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

A logarithm answers "what power?": $\log_{a} b = c$ means $a^{c} = b$ . In P2 you use the laws of logarithms to combine and simplify, solve $a^{x} = b$ by taking logs, and solve equations that contain logs, rejecting any answer that makes a log's argument zero or negative. Natural logarithms and $e^{x}$ come in P3.

P2 · Pure Mathematics 2
Calculator allowed
3 worked examples
35 practice questions
Tested in 4 of the last 4 papers

Practise P2 logarithms and exponentials →

Notes
Formulas
Worked examples
Mistakes
FAQ

P2 Logarithms and Exponentials notes

Exponential graphs

$y = a^{x}$ with $a > 1$ is increasing, passes through $(0, 1)$ and has the asymptote $y = 0$ ; it is never negative. With $0 < a < 1$ it decreases.

What a logarithm is

$\log_{a} b = c$ and $a^{c} = b$ say the same thing ( $a > 0$ , $a \neq 1$ , $b > 0$ ). So $\log_{2} 64 = 6$ and $\log_{5} \frac{1}{25} = - 2$ . Also $\log_{a} 1 = 0$ and $\log_{a} a = 1$ .

The laws of logarithms

$\log_{a} x y = \log_{a} x + \log_{a} y$ , $\log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y$ and $\log_{a} x^{k} = k \log_{a} x$ . Use them to write several logs as one: $2 \log_{3} 6 - \log_{3} 4 = \log_{3} 9 = 2$ .

Solving exponential and log equations

Unknown in the power: take logs of both sides. $5^{x} = 40$ gives $x = \frac{\log 40}{\log 5} \approx 2.29$ . Logs in the equation: combine into one log, then undo it. $\log_{2} (x + 2) + \log_{2} x = 3$ gives $x (x + 2) = 8$ , so $x = 2$ ; reject $x = - 4$ , because $\log_{2} (- 4)$ does not exist.

Change of base

$\log_{a} x = \frac{\log_{b} x}{\log_{b} a}$ . It lets you evaluate any log on a calculator and simplify mixed bases: $\log_{8} 2 = \frac{\log_{2} 2}{\log_{2} 8} = \frac{1}{3}$ .

Key formulas

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 P2 formulas on one page (free PDF) →

P2 Logarithms and Exponentials 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

Solve the equation $\log_{2} x + \log_{x} 2 = 2.5$ for $x > 0, x \neq 1$ .

Mark scheme

M1 for applying the reciprocal rule: $\log_{x} 2 = \frac{1}{\log_{2} x}$ .
M1 for making the substitution $u = \log_{2} x$ to form $u + \frac{1}{u} = 2.5$ .
M1 for multiplying by $2 u$ to form the quadratic $2 u^{2} - 5 u + 2 = 0$ and solving it.
M1 for finding the roots $u = 2$ and $u = \frac{1}{2}$ .
A1 for converting back to $x$ and finding both solutions: $x = 4$ and $x = \sqrt{2}$ .

5 steps · 1 to fill in

Worked example 2 · 4 marks · hard

Given that $\log_{2} x = p$ and $\log_{2} y = q$ , express $\log_{2} (8 x^{2} y^{3})$ in terms of $p$ and $q$ .

Mark scheme

M1 for splitting the logarithm into a sum: $\log_{2} 8 + \log_{2} x^{2} + \log_{2} y^{3}$ .
B1 for evaluating $\log_{2} 8 = 3$ .
M1 for applying the power law: $2 \log_{2} x + 3 \log_{2} y$ .
A1 for the final expression $3 + 2 p + 3 q$ .

4 steps

Worked example 3 · 4 marks · hard

Solve the equation $3^{x + 2} = 7^{x - 1}$ , giving your answer to 3 decimal places.

Mark scheme

M1 for taking logarithms of both sides: $(x + 2) \log 3 = (x - 1) \log 7$ .
M1 for expanding the brackets: $x \log 3 + 2 \log 3 = x \log 7 - \log 7$ .
M1 for isolating $x$ terms on one side: $x (\log 7 - \log 3) = 2 \log 3 + \log 7$ .
A1 for evaluating $x = \frac{2 \log 3 + \log 7}{\log 7 - \log 3} = 4.890$ (3 d.p.).

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 P2 logarithms and exponentials mistakes

Splitting the log of a sum. $\log (x + y)$ is not $\log x + \log y$ . Only products and quotients split.
Keeping an impossible root. After solving, check every answer in the original equation: a log of zero or of a negative number means the answer must be rejected, and you should say why.
Rounding too early. Keep logs exact or at full calculator accuracy until the last line, then round as the question asks.
The power law on the wrong thing. $\log_{a} x^{k} = k \log_{a} x$ , but $(\log_{a} x)^{2}$ is not $2 \log_{a} x$ .
Taking logs before isolating the power. In $2 \times 3^{x} = 15$ , divide by 2 first: $3^{x} = 7.5$ , so $x = \frac{\ln 7.5}{\ln 3} \approx 1.834$ . Taking logs straight away works, but gives $\log 2 + x \log 3 = \log 15$ , a longer route with more room for slips; and $\log (2 \times 3^{x})$ is not $x \log 6$ .

Every P2 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.

P2 Logarithms and Exponentials FAQ

Are the laws of logarithms in the IAL formula booklet?

On our IAL formula sheet the change of base formula is marked In the formula booklet and the three laws of logarithms are marked Learn it, so learn the laws by heart.

Do I need natural logarithms in P2?

No. ln x and e^x are P3 content. In P2 you work with logarithms to a general base a, and your calculator's log key for base 10.

Is a calculator allowed in IAL P2?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P2 included. Method marks are for the working you write down, so show it.

How long is the IAL P2 exam?

Each IAL maths unit, P2 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does logarithms and exponentials come up in P2?

In our mapping of the 4 Pearson P2 papers from June 2024 to June 2025, logarithms and exponentials was tested in 4 of them (6 questions in all). Any topic can be examined in any series.

Next steps

More P2: Sequences and Series · Circles · Further Trigonometry · Algebraic Methods · Proof · The Binomial Expansion · Differentiation (stationary points) · Integration (definite integrals and area)

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Edexcel IAL P2 Logarithms and Exponentials: exam questions and answers, notes and worked examples

Try it first

P2 Logarithms and Exponentials notes

Exponential graphs

What a logarithm is

The laws of logarithms

Solving exponential and log equations

Change of base

Key formulas

P2 Logarithms and Exponentials worked examples

Worked example 1 · 5 marks · hard

Mark scheme

Worked example 2 · 4 marks · hard

Mark scheme

Worked example 3 · 4 marks · hard

Mark scheme

Common P2 logarithms and exponentials mistakes

P2 Logarithms and Exponentials FAQ

Are the laws of logarithms in the IAL formula booklet?

Do I need natural logarithms in P2?

Is a calculator allowed in IAL P2?

How long is the IAL P2 exam?

How often does logarithms and exponentials come up in P2?

Next steps

Your P2 revision kit