What are the key differences between logarithmic and exponential functions in the Cambridge IGCSE Additional Mathematics syllabus?

In the Cambridge IGCSE Additional Mathematics syllabus, exponential functions are of the form y = a^x, where 'a' is a positive constant. These functions model growth or decay processes. Logarithmic functions, on the other hand, are the inverse of exponential functions and are of the form y = log_a(x), where 'a' is the base of the logarithm. They are used to solve equations involving exponential growth or decay.

How do you solve exponential equations in the context of Cambridge IGCSE Additional Mathematics?

To solve exponential equations in Cambridge IGCSE Additional Mathematics, 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. It's important to understand the properties of logarithms to manipulate and solve these equations effectively.

What is the significance of the natural logarithm in exponential functions?

The natural logarithm, denoted as ln(x), is the logarithm to the base 'e', where 'e' is approximately 2.718. It is significant in exponential functions because it simplifies the differentiation and integration of exponential functions. In the Cambridge IGCSE Additional Mathematics curriculum, understanding the natural logarithm is crucial for solving problems involving continuous growth or decay.

How can you graph logarithmic and exponential functions for Cambridge IGCSE Additional Mathematics?

To graph exponential functions, you plot points for various values of x and connect them to show the curve, which typically rises or falls rapidly. For logarithmic functions, you plot points for different values of x and connect them to show a curve that increases slowly and passes through the point (1,0). Understanding the asymptotic behavior and intercepts is key for accurate graphing in Cambridge IGCSE Additional Mathematics.

What are some real-world applications of logarithmic and exponential functions taught in Cambridge IGCSE Additional Mathematics?

Logarithmic and exponential functions have numerous real-world applications, which are covered in the Cambridge IGCSE Additional Mathematics syllabus. Exponential functions model population growth, radioactive decay, and compound interest. Logarithmic functions are used in measuring the intensity of earthquakes (Richter scale) and sound (decibels). Understanding these applications helps students appreciate the practical relevance of these mathematical concepts.

Why is "Treating $\log(x + y)$ as $\log x + \log y$" a common mistake in Logarithmic and Exponential Functions?

Why it happens: Confusion with the product law. How to avoid it: Log laws apply only to PRODUCTS, QUOTIENTS, POWERS — NOT to sums. (x + y) x + y. Source: 0606 Examiner Reports 2022-2024

Why is "Mixing up the base when solving" a common mistake in Logarithmic and Exponential Functions?

Why it happens: Multiple bases in play. How to avoid it: Always specify the base. Use the same base on both sides when solving. has base e; usually means base 10 in calculator contexts but base e in advanced maths.

Why is "Failing to bring exponents down" a common mistake in Logarithmic and Exponential Functions?

Why it happens: Skipping the power-law step. How to avoid it: When you have \log_a x^k, immediately bring the k down: k \log_a x. Especially key in solving exponential equations.

Logarithmic and Exponential FunctionsCambridge IGCSE Additional Mathematics (0606)

Logarithmic and Exponential Functions

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Logarithmic and Exponential Functions Study Notes — Cambridge IGCSE Additional Mathematics 0606 (2025-2027 syllabus)

Exponentials $a^x$ and $e^x$ . Logarithms as inverses. Three log laws. Solving exponential and logarithmic equations. Graphs of $e^x$ , $\ln x$ , and transformations.

What you’ll learn

Mapped to the Cambridge IGCSE 0606 syllabus (2025-2027).

1.7.1 — Define exponential and logarithm.
1.7.2 — Apply the three log laws.
1.7.3 — Solve exponential equations using logs.
1.7.4 — Solve logarithmic equations.
1.7.5 — Sketch $e^x$ and $\ln x$ and their transformations.

Exponentials and logarithms

Exponentials grow / decay rapidly; logarithms reverse them.

Exponential function. $f(x) = a^x$ where $a > 0$ , $a \ne 1$ .

Always positive.
Passes through $(0, 1)$ since $a^0 = 1$ .
Increasing if $a > 1$ ; decreasing if $0 < a < 1$ .

Natural exponential. $f(x) = e^x$ where $e \approx 2.71828$ . Special because $\frac{d}{dx} e^x = e^x$ .

Logarithm. $\log_a y = x$ means $a^x = y$ . The logarithm UNDOES exponentiation.

Natural logarithm. $\ln x = \log_e x$ .

Inverse relationships. $e^{\ln x} = x \text{ for } x > 0, \quad \ln(e^x) = x \text{ for all } x.$

Cambridge tip. Memorise: $\log_a 1 = 0$ and $\log_a a = 1$ . These follow directly from $a^0 = 1$ and $a^1 = a$ .

Exponential: $a^x$ , always positive.
Log: inverse of exp.
$e \approx 2.718$ ; $\ln$ = base $e$ .

The log laws

Three rules for combining logarithms.

Three core laws (assume same base $a$ , positive arguments).

Product: $\log_a(xy) = \log_a x + \log_a y$ .

Quotient: $\log_a(x/y) = \log_a x - \log_a y$ .

Power: $\log_a(x^k) = k \log_a x$ .

Crucial: NO law for sums. $\log(x + y) \ne \log x + \log y$ . Don't break sums apart inside a logarithm.

Worked walkthrough. Simplify $\log_a 12 + \log_a 3 - 2 \log_a 6$ .

$2 \log_a 6 = \log_a 36$ (power law).
$\log_a 12 + \log_a 3 = \log_a 36$ (product law).
$\log_a 36 - \log_a 36 = \log_a 1 = 0$ .

Cambridge tip. When asked to combine logs to a single expression, use the laws in order: power → product → quotient.

Product, quotient, power — three laws.
NO sum law inside a log.
Apply power law first to bring exponents in front.

See the full worked example for logarithmic and exponential functions →

Solving exponential and log equations

Take logs of both sides for exponentials; convert log equations into exp form.

Exponential equations. Take logs of both sides; use the power law to bring the unknown out of the exponent.

Worked walkthrough. $5^x = 100$ .

$\ln(5^x) = \ln 100$ .
$x \ln 5 = \ln 100$ .
$x = \dfrac{\ln 100}{\ln 5} \approx 2.86$ .

Mixed equations. $2 \cdot 3^{x+1} = 5^x$ .

$\ln 2 + (x + 1) \ln 3 = x \ln 5$ .
$\ln 2 + x \ln 3 + \ln 3 = x \ln 5$ .
$\ln 6 = x(\ln 5 - \ln 3) = x \ln(5/3)$ .
$x = \dfrac{\ln 6}{\ln(5/3)}$ .

Logarithmic equations. Convert to exponential form: $\log_a y = x \Rightarrow y = a^x$ .

Worked walkthrough. $\log_2(x - 1) = 3$ .

$x - 1 = 2^3 = 8$ .
$x = 9$ .

Cambridge tip. When you're stuck, take logs of both sides if the unknown is in the exponent; convert to exponential form if the equation has $\log$ .

Exp equation: take logs.
Log equation: convert to exp.
Use power law to bring unknown out of exponent.

See the full worked example for logarithmic and exponential functions →

Applications: growth and decay

Real-world models: $A = A_0 e^{kt}$ for growth; $A = A_0 e^{-kt}$ for decay.

Population growth. $P(t) = P_0 e^{rt}$ where $r > 0$ is the growth rate.

Radioactive decay. $N(t) = N_0 e^{-\lambda t}$ where $\lambda > 0$ is the decay constant.

Compound interest (continuous). $A = P e^{rt}$ .

Solving for time. Take natural log: $\ln\left(\frac{A}{A_0}\right) = kt \Rightarrow t = \frac{1}{k} \ln\left(\frac{A}{A_0}\right).$

Worked walkthrough. Substance decays as $N = N_0 e^{-0.05 t}$ . Find time to reach 25% of original.

$0.25 = e^{-0.05 t}$ .
$\ln 0.25 = -0.05 t$ .
$t = \dfrac{\ln 0.25}{-0.05} \approx 27.7$ years.

Cambridge tip. Clearly identify $A_0$ , $A$ , and $k$ before solving. Decay has NEGATIVE exponent.

Growth: $A_0 e^{kt}$ with $k > 0$ .
Decay: $A_0 e^{-kt}$ with $k > 0$ .
Solve for time by taking $\ln$ .

See the full worked example for logarithmic and exponential functions →

How it’s examined

Logarithms and exponentials appear on every Paper. Most-tested: simplify using log laws (5 marks), solve exponential equation (5-7 marks), exponential growth/decay (5 marks).

Sources: Cambridge IGCSE Additional Mathematics 0606 syllabus (2025-2027); 0606 Examiner Reports 2022-2024; 0606/12 May/Jun 2024 question paper and mark scheme. Last reviewed 2026-05-09.

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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.

1Apply log laws (5 marks)
Extended• Adapted from 0606/12 May/Jun 2024 Q7• logarithms
Question
Express $\log_a 12 + \log_a 3 - 2 \log_a 6$ as a single logarithm. (5 marks)
Step-by-step solution
1. Step 1
  Power law (1 mark). $2 \log_a 6 = \log_a 6^2 = \log_a 36$ .
2. Step 2
  Product law (2 marks). $\log_a 12 + \log_a 3 = \log_a (12 \cdot 3) = \log_a 36$ .
3. Step 3
  Quotient law (2 marks). $\log_a 36 - \log_a 36 = \log_a \dfrac{36}{36} = \log_a 1 = 0$ .
Answer
$\log_a 1 = 0$ .
2Solve an exponential equation (6 marks)
Extended• exponential
Question
Solve $2 \cdot 3^{x+1} = 5^x$ . Give your answer correct to 3 significant figures. (6 marks)
Step-by-step solution
1. Step 1
  Take logs of both sides (1 mark). $\ln(2 \cdot 3^{x+1}) = \ln(5^x)$ .
2. Step 2
  Apply log laws (2 marks). $\ln 2 + (x+1) \ln 3 = x \ln 5$ .
3. Step 3
  Expand (1 mark). $\ln 2 + x \ln 3 + \ln 3 = x \ln 5$ .
4. Step 4
  Collect $x$ terms (1 mark). $\ln 2 + \ln 3 = x \ln 5 - x \ln 3 = x(\ln 5 - \ln 3)$ .
5. Step 5
  Solve (1 mark). $x = \dfrac{\ln 6}{\ln(5/3)} = \dfrac{1.7918}{0.5108} \approx 3.51$ (3 s.f.).
Answer
$x \approx 3.51$ .
3Exponential decay (5 marks)
Extended• exponential, application
Question
A radioactive substance decays according to $N = N_0 e^{-0.05 t}$ , where $t$ is in years. Find the time for the substance to decay to 25% of its original amount. (5 marks)
Step-by-step solution
1. Step 1
  Set up the equation (1 mark). $0.25 N_0 = N_0 e^{-0.05 t}$ , so $0.25 = e^{-0.05 t}$ .
2. Step 2
  Take natural log (2 marks). $\ln 0.25 = -0.05 t$ .
3. Step 3
  Solve (2 marks). $t = \dfrac{\ln 0.25}{-0.05} = \dfrac{-1.3863}{-0.05} \approx 27.7$ years.
Answer
$t \approx 27.7$ years.

Key Formulae — Logarithmic and Exponential Functions

The formulae you need to memorise for logarithmic and exponential functions on the Cambridge IGCSE 0606 paper, with every variable defined in plain English and a note on when to use it.

Log product law
$\log_a(xy) = \log_a x + \log_a y$
$a$
Base, $a > 0, a \ne 1$
$x, y$
Positive reals
When to use
Combining or splitting logarithms of products.
Log quotient law
$\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$
$a, x, y$
Base and arguments
When to use
Combining or splitting logarithms of quotients.
Log power law
$\log_a(x^k) = k \log_a x$
$a, x$
Base and argument
$k$
Any real number
When to use
Bringing exponents in front of a logarithm.
Change of base
$\log_a x = \frac{\log_b x}{\log_b a}$
$a, b$
Bases
$x$
Positive real
When to use
Converting between log bases (e.g. evaluating $\log_3 7$ via natural log).
Natural log
$\ln x = \log_e x \quad \text{and} \quad e^{\ln x} = x \text{ for } x > 0$
$e$
Euler's number, approximately 2.71828
When to use
Solving exponential equations involving $e^x$ .

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 0606 sitting.

Exponential function
Examiner keyword
A function of the form $f(x) = a^x$ where $a > 0, a \ne 1$ . Always positive; passes through $(0, 1)$ .
Logarithm
Examiner keyword
$\log_a y = x$ means $a^x = y$ . Logarithm is the inverse of exponentiation.
Natural logarithm $\ln x$
Examiner keyword
Logarithm with base $e$ . $\ln x = \log_e x$ .
Euler's number $e$
Examiner keyword
An irrational mathematical constant, approximately $2.71828$ . Base of the natural logarithm.

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 IGCSE 0606 examiner reports and mark schemes — and how to avoid them.

✕Treating $\log(x + y)$ as $\log x + \log y$
0606 Examiner Reports 2022-2024
Why it happens
Confusion with the product law.
How to avoid it
Log laws apply only to PRODUCTS, QUOTIENTS, POWERS — NOT to sums. $\log(x + y) \ne \log x + \log y$ .
✕Mixing up the base when solving
Why it happens
Multiple bases in play.
How to avoid it
Always specify the base. Use the same base on both sides when solving. $\ln$ has base $e$ ; $\log$ usually means base 10 in calculator contexts but base $e$ in advanced maths.
✕Failing to bring exponents down
Why it happens
Skipping the power-law step.
How to avoid it
When you have $\log_a x^k$ , immediately bring the $k$ down: $k \log_a x$ . Especially key in solving exponential equations.

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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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 Notes

What you’ll learn

How it’s examined

1Apply log laws (5 marks)

2Solve an exponential equation (6 marks)

3Exponential decay (5 marks)

Log product law

Log quotient law

Log power law

Change of base

Natural log

Exponential function

Logarithm

Natural logarithm ln⁡x\ln xlnx

Euler's number eee

✕Treating log⁡(x+y)\log(x + y)log(x+y) as log⁡x+log⁡y\log x + \log ylogx+logy

✕Mixing up the base when solving

✕Failing to bring exponents down

Practice questions

Past paper style quiz

Assess your understanding

Logarithmic and Exponential Functions — frequently asked questions

Natural logarithm $\ln x$

Euler's number $e$

✕Treating $\log(x + y)$ as $\log x + \log y$