What are the key differences between logarithmic and exponential functions in Pure Mathematics 2?

In Cambridge International A Levels Pure Mathematics 2, exponential functions are expressed in the form y = a^x, where 'a' is a positive constant, and they model growth or decay processes. Logarithmic functions, on the other hand, are the inverse of exponential functions and are expressed as y = log_a(x), where 'a' is the base. While exponential functions grow rapidly, logarithmic functions increase slowly and are used to solve equations involving exponentials.

How do you solve exponential equations in the context of Cambridge International A Levels?

To solve exponential equations in Pure Mathematics 2, you often use logarithms. For example, if you have an equation like a^x = b, you can take the logarithm of both sides to get x = log_a(b). This allows you to solve for x by using properties of logarithms. Understanding the relationship between exponential and logarithmic functions is crucial for solving such equations.

What are the applications of logarithmic and exponential functions in real-world scenarios?

Logarithmic and exponential functions have numerous applications in real-world scenarios, which are often explored in Cambridge International A Levels Mathematics. Exponential functions model phenomena such as population growth, radioactive decay, and compound interest. Logarithmic functions are used in measuring the intensity of earthquakes (Richter scale), sound (decibels), and in various scientific calculations where scaling is necessary.

How do you differentiate logarithmic and exponential functions in Pure Mathematics 2?

In Cambridge International A Levels Pure Mathematics 2, differentiating exponential functions like y = e^x results in dy/dx = e^x. For logarithmic functions, such as y = ln(x), the derivative is dy/dx = 1/x. These derivatives are fundamental in solving calculus problems involving growth rates and changes in quantities over time.

What are the properties of logarithms that are essential for solving problems in Pure Mathematics 2?

Key properties of logarithms essential for Cambridge International A Levels include the product rule (log_a(xy) = log_a(x) + log_a(y)), the quotient rule (log_a(x/y) = log_a(x) - log_a(y)), and the power rule (log_a(x^n) = n*log_a(x)). These properties simplify complex logarithmic expressions and are crucial for solving logarithmic equations and inequalities.

Why is "$\log(x + y) \ne \log x + \log y$" a common mistake in Logarithmic and exponential functions?

Why it happens: Confusion with product rule. How to avoid it: Product: (xy) = x + y. Sum: NO simplification. Source: 9709 Examiner Reports 2022-2024

Why is "Taking log of zero or negative" a common mistake in Logarithmic and exponential functions?

Why it happens: Not checking domain. How to avoid it: x defined only for x > 0. Check final answer doesn't require log of non-positive. Source: 9709 Examiner Reports 2022-2024

Pure Mathematics 2Cambridge International A Levels Mathematics (9709)

Logarithmic and exponential functions

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Logarithmic and Exponential Functions P2 Study Notes — Cambridge International A Level Mathematics 9709 (2024-2026 syllabus)

Log laws. $e$ and $\ln$ . Exponential equations. Linearisation of $y = ax^n$ and $y = ab^x$ .

What you’ll learn

Mapped to the Cambridge IGCSE 9709 syllabus (2024-2026).

P2.2.1 — Use log laws.
P2.2.2 — Solve exponential and log equations.
P2.2.3 — Linearise power and exponential laws.

Log laws

Product, quotient, power.

Logarithm. $\log_a x = y$ means $a^y = x$ . Inverse of exponentiation.

Three laws.

Product: $\log_a(xy) = \log_a x + \log_a y$ .
Quotient: $\log_a(x/y) = \log_a x - \log_a y$ .
Power: $\log_a(x^n) = n \log_a x$ .

Special values. $\log_a 1 = 0$ . $\log_a a = 1$ .

Change of base. $\log_a x = \frac{\log_b x}{\log_b a}$ . Useful when calculator only has $\log_{10}$ or $\ln$ .

Example. $\log_a 8 + 2\log_a 3 - \log_a 6$

$2\log_a 3 = \log_a 9$ .
$\log_a 8 + \log_a 9 - \log_a 6 = \log_a \frac{72}{6} = \log_a 12$ .

Cambridge tip. $\log(x + y) \ne \log x + \log y$ . Memorise!

Product, quotient, power.
$\log a + \log b = \log ab$ .
Sum is NOT log of sum.

See the full worked example for logarithmic and exponential functions →

$e$ and natural log

Base $e \approx 2.718$ .

$e$ is a mathematical constant, $\approx 2.718$ . Special because $\frac{d}{dx}e^x = e^x$ .

$\ln x = \log_e x$ . Natural log. The inverse of $e^x$ .

Properties.

$e^{\ln x} = x$ (for $x > 0$ ).
$\ln e^x = x$ .
$\ln 1 = 0$ . $\ln e = 1$ .
All log laws apply to $\ln$ .

Solving $a^{f(x)} = b$ .

Take $\ln$ of both sides.
Use power law: $f(x) \ln a = \ln b$ .
Solve for $x$ .

Example. $3^{2x+1} = 17$ .

$\ln 3^{2x+1} = \ln 17$ .
$(2x+1)\ln 3 = \ln 17$ .
$2x + 1 = \frac{\ln 17}{\ln 3} \approx 2.579$ .
$x \approx 0.790$ .

Cambridge tip. Keep exact form ( $\frac{\ln 17}{\ln 3}$ ) until final answer.

$e \approx 2.718$ .
$\ln$ is log base $e$ .
Take $\ln$ to solve exponential.

See the full worked example for logarithmic and exponential functions →

Linearisation

Power and exponential laws.

Power law: $y = ax^n$ . Take logs: $\log y = \log a + n \log x$ .

Plot $\log y$ vs $\log x$ : straight line, gradient $n$ , intercept $\log a$ .

Exponential law: $y = ab^x$ . Take logs: $\log y = \log a + x \log b$ .

Plot $\log y$ vs $x$ : straight line, gradient $\log b$ , intercept $\log a$ .

Identify which to use.

$y = ax^n$ → log-log plot.
$y = ab^x$ → semi-log plot ( $\log y$ vs $x$ ).

Example. Data follows $y = ax^n$ .

Linearise: $\log y = n \log x + \log a$ .
Plot: $\log y$ on vertical, $\log x$ on horizontal.
Gradient gives $n$ directly.
$\log a$ is intercept; $a = 10^{\text{intercept}}$ .

Cambridge tip. Always state CLEARLY what you plot, what the gradient and intercept represent.

Power law: log-log plot.
Exponential: semi-log plot.
Gradient and intercept = parameters.

See the full worked example for logarithmic and exponential functions →

How it’s examined

Logs/exponentials appear every P2 — usually a 7-10 mark question. Most-tested: exponential equation (5 marks), log simplification (5 marks), linearisation (7-10 marks).

Sources: Cambridge International A Level Mathematics 9709 syllabus (2024-2026); 9709 Examiner Reports 2022-2024; 9709/22 May/Jun 2024 question paper and mark scheme. Last reviewed 2026-05-11.

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Step-by-step worked examples — Logarithmic and exponential functions

Step-by-step solutions to past-paper-style questions on logarithmic and exponential functions, written exactly the way a tutor would explain them at the board.

1Solving an exponential equation (5 marks)
Extended• Adapted from 9709/22 May/Jun 2024• exponential
Question
Solve $3^{2x+1} = 17$ giving your answer to 3 sf. (5 marks)
Step-by-step solution
1. Step 1
  Take logs of both sides.
  $\ln 3^{2x+1} = \ln 17$
2. Step 2
  Use log power rule.
  $(2x+1) \ln 3 = \ln 17$
3. Step 3
  Solve for $x$ .
  $2x + 1 = \dfrac{\ln 17}{\ln 3} \approx 2.579$
4. Step 4
  Isolate $x$ .
  $x \approx \dfrac{2.579 - 1}{2} \approx 0.790$
Answer
$x \approx 0.790$ (3 sf).
2Log laws (5 marks)
Extended• log laws
Question
Express $\log_a 8 + 2\log_a 3 - \log_a 6$ as a single log. (5 marks)
Step-by-step solution
1. Step 1
  Power law on middle term.
  $2\log_a 3 = \log_a 9$
2. Step 2
  Sum and difference.
  $\log_a 8 + \log_a 9 - \log_a 6 = \log_a \dfrac{8 \times 9}{6}$
3. Step 3
  Simplify.
  $= \log_a 12$
Answer
$\log_a 12$ .
3Linearisation of $y = ax^n$ (7 marks)
Extended• Adapted from 9709/22 Oct/Nov 2024• linearisation
Question
Data follows $y = ax^n$ . By taking logs of both sides, show how to plot a straight-line graph. State what the gradient and intercept represent. (7 marks)
Step-by-step solution
1. Step 1
  Take logs.
  $\log y = \log(a x^n) = \log a + n \log x$
2. Step 2
  Compare to $Y = mX + c$ . $Y = \log y$ , $X = \log x$ .
3. Step 3
  Plot $\log y$ against $\log x$ . Gives straight line.
4. Step 4
  Gradient $= n$ . Intercept on $\log y$ -axis $= \log a$ .
5. Step 5
  To find $a$ . $a = 10^{\text{intercept}}$ (or $e^{\text{intercept}}$ if using ln).
Answer
$\log y = n \log x + \log a$ . Plot $\log y$ vs $\log x$ . Gradient = $n$ , intercept = $\log a$ .

Key Formulae — Logarithmic and exponential functions

The formulae you need to memorise for logarithmic and exponential functions on the Cambridge International A Level 9709 paper, with every variable defined in plain English and a note on when to use it.

Product rule
$\log_a(xy) = \log_a x + \log_a y$
When to use
Combine or split logs of products.
Quotient rule
$\log_a(x/y) = \log_a x - \log_a y$
When to use
Combine or split logs of quotients.
Power rule
$\log_a(x^n) = n \log_a x$
When to use
Bring exponents outside logs.
Change of base
$\log_a x = \dfrac{\log_b x}{\log_b a}$
When to use
Evaluate $\log_a$ using a different base (usually $\ln$ or $\log_{10}$ ).
$e$ and $\ln$ inverses
$e^{\ln x} = x, \quad \ln e^x = x$
When to use
Solving equations involving $e^x$ or $\ln x$ . They cancel each other.

Key Definitions and Keywords — Logarithmic and exponential functions

Definitions to memorise and the exact keywords mark schemes credit for logarithmic and exponential functions answers — sharpened from recent examiner reports for the 2026 Cambridge International A Level 9709 sitting.

Logarithm
Examiner keyword
$\log_a x = y$ means $a^y = x$ . Inverse of exponentiation.
Natural log $\ln x$
Examiner keyword
Logarithm to base $e$ . $\ln x = \log_e x$ . $e \approx 2.718$ .
Exponential function $e^x$
Examiner keyword
Function with base $e$ . Always positive. Inverse of $\ln x$ . $e^0 = 1$ , $e^1 = e$ .

Common Mistakes and Misconceptions — Logarithmic and exponential functions

The traps other students keep falling into on logarithmic and exponential functions questions — taken from recent Cambridge International A Level 9709 examiner reports and mark schemes — and how to avoid them.

✕ $\log(x + y) \ne \log x + \log y$
9709 Examiner Reports 2022-2024
Why it happens
Confusion with product rule.
How to avoid it
Product: $\log(xy) = \log x + \log y$ . Sum: NO simplification.
✕Taking log of zero or negative
9709 Examiner Reports 2022-2024
Why it happens
Not checking domain.
How to avoid it
$\log x$ defined only for $x > 0$ . Check final answer doesn't require log of non-positive.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Video lesson

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Logarithmic and exponential functions — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Logarithmic and exponential functions

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Solving an exponential equation (5 marks)

2Log laws (5 marks)

3Linearisation of y=axny = ax^ny=axn (7 marks)

Product rule

Quotient rule

Power rule

Change of base

eee and ln⁡\lnln inverses

Logarithm

Natural log ln⁡x\ln xlnx

Exponential function exe^xex

✕log⁡(x+y)≠log⁡x+log⁡y\log(x + y) \ne \log x + \log ylog(x+y)=logx+logy

✕Taking log of zero or negative

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Logarithmic and exponential functions — frequently asked questions

3Linearisation of $y = ax^n$ (7 marks)

$e$ and $\ln$ inverses

Natural log $\ln x$

Exponential function $e^x$

✕ $\log(x + y) \ne \log x + \log y$