What does this module cover in N(A)-Level A-Maths 4051?

It covers three outcomes in the Algebra strand of syllabus 4051: defining a logarithm as the inverse of a power and using the product, quotient and power laws of logarithms; sketching exponential and logarithmic graphs and recognising them as reflections of each other; and solving exponential and logarithmic equations. The laws underpin both the graphs and the equation solving.

A logarithm answers the question 'to what power must the base be raised to give this number?'. The statement $\log_a x = y$ means exactly the same as $a^y = x$, so a logarithm is the inverse of a power. For example $\log_2 8 = 3$ because $2^3 = 8$.

What are the three laws of logarithms?

The product law is $\log_a (xy) = \log_a x + \log_a y$; the quotient law is $\log_a \dfrac{x}{y} = \log_a x - \log_a y$; and the power law is $\log_a (x^n) = n \log_a x$. These let you combine several logs into one or break one into parts when simplifying or solving.

How are the exponential and logarithmic graphs related?

The graphs of $y = a^x$ and $y = \log_a x$ are reflections of each other in the line $y = x$, because the functions are inverses. The exponential curve passes through $(0, 1)$ with the $x$-axis as an asymptote, while the logarithmic curve passes through $(1, 0)$ with the $y$-axis as an asymptote.

How is the Algebra strand assessed in syllabus 4051?

Both Paper 1 and Paper 2 (each 1 hour 45 minutes, 70 marks, weighted 50 percent) draw on the whole syllabus, and candidates answer all questions. Logarithm and exponential questions usually test the laws and equation solving, so practise moving between index form and log form fluently.

SingaporeAdditional Mathematics

Logarithmic and exponential functions in N(A)-Level Additional Mathematics (4051): the definition and laws of logarithms, the graphs of exponential and logarithmic functions and their reflection symmetry, and solving exponential and logarithmic equations

An N(A)-Level Additional Mathematics (4051) overview of logarithmic and exponential functions in the Algebra strand. The definition of a logarithm as the inverse of a power, the product, quotient and power laws, the shapes and symmetry of the graphs, and how to solve exponential and logarithmic equations, with links to every dot point.

Generated by Claude Opus 4.86 min readSEAB-4051 AlgebraUpdated 2026-06-13

Reviewed by: AI editorial process; not yet individually human-reviewed

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Powers and their inverses
The definition and laws of logarithms
The graphs and their symmetry
Solving exponential and logarithmic equations
How this module is examined
Check your knowledge

Powers and their inverses

Logarithmic and exponential functions are a paired topic in the Algebra strand of N(A)-Level Additional Mathematics (SEAB 4051). An exponential function raises a fixed base to a variable power; a logarithm is its inverse, asking which power produced a given number. Once you see logarithms as undoing powers, the laws, the graphs and the equation solving all follow.

This overview links three dot points: the laws of logarithms, the graphs of exponential and logarithmic functions, and solving exponential and logarithmic equations. Each has its own worked answers and practice.

The definition and laws of logarithms

The laws of logarithms outcome starts from the definition $\log_a x = y \iff a^y = x$ , then gives three laws. The product law is $\log_a (xy) = \log_a x + \log_a y$ , the quotient law is $\log_a \dfrac{x}{y} = \log_a x - \log_a y$ , and the power law is $\log_a (x^n) = n \log_a x$ . These combine or split logarithms.

The graphs and their symmetry

The graphs of exponential and logarithmic functions outcome covers their shapes. The exponential curve $y = a^x$ (with $a$ greater than $1$ ) rises steeply, passes through $(0, 1)$ , and has the $x$ -axis as a horizontal asymptote. The logarithmic curve $y = \log_a x$ passes through $(1, 0)$ and has the $y$ -axis as a vertical asymptote. Being inverses, the two graphs are reflections of each other in the line $y = x$ .

Solving exponential and logarithmic equations

The exponential and logarithmic equations outcome puts the laws to work. To solve an exponential equation where the unknown is in the exponent, take logarithms of both sides and use the power law to bring the exponent down. To solve a logarithmic equation, combine the logs with the laws and then switch to index form.

How this module is examined

Both papers, all questions. Paper 1 and Paper 2 (each 1 hour 45 minutes, 70 marks, 50 percent) cover the full syllabus, and you answer every question.
Give answers to 3 significant figures. Unless told otherwise, round a non-exact logarithm answer to 3 significant figures, as the syllabus requires.
Check solutions of log equations. A logarithm is only defined for a positive argument, so reject any solution that makes the inside of a log zero or negative.

Logarithmic and exponential functions sit in the Algebra strand of N(A)-Level A-Maths (SEAB 4051). A logarithm is the inverse of a power: $\log_a x = y \iff a^y = x$ . The three laws are the product law $\log_a (xy) = \log_a x + \log_a y$ , the quotient law $\log_a \dfrac{x}{y} = \log_a x - \log_a y$ , and the power law $\log_a (x^n) = n \log_a x$ . The graphs of $y = a^x$ and $y = \log_a x$ are reflections in the line $y = x$ . To solve an exponential equation, take logs and use the power law; to solve a log equation, combine logs then switch to index form, rejecting non-positive arguments. Both papers draw on the whole syllabus.

Check your knowledge

Short questions across the three outcomes. Work them with full method, then check the solutions.

Evaluate $\log_2 16$ . (1 mark)
Write $\log_a x + \log_a y$ as a single logarithm. (1 mark)
Solve $\log_3 x = 4$ . (2 marks)
Solve $5^x = 100$ , giving $x$ to 3 significant figures. (3 marks)

Solutions

The four questions cover evaluating a logarithm, applying the product law, solving a log equation in index form, and solving an exponential equation.

Step 1: Evaluate $\log_2 16$

The statement $\log_2 16 = y$ means $2^y = 16$ . Since $2^4 = 16$ , the answer follows directly without a calculator:

\log_2 16 = 4.

Final answer (Q1): $\log_2 16 = 4$ .

Step 2: Write $\log_a x + \log_a y$ as a single logarithm

The product law states that the logarithm of a product equals the sum of the logarithms. Applying it in reverse combines two separate logs into one:

\log_a x + \log_a y = \log_a (xy).

Final answer (Q2): $\log_a (xy)$ .

Step 3: Solve $\log_3 x = 4$

Convert from logarithm form to index form using the definition $\log_a x = y \iff a^y = x$ . Here the base is $3$ and the logarithm equals $4$ :

x = 3^4 = 81.

Final answer (Q3): $x = 81$ .

Step 4: Solve $5^x = 100$ to 3 significant figures

Take the common logarithm of both sides to bring the unknown exponent down using the power law, then divide:

x \log 5 = \log 100 \implies x = \frac{\log 100}{\log 5} = \frac{2}{0.699} = 2.86 \text{ (to 3 s.f.)}.

Final answer (Q4): $x = 2.86$ (to 3 significant figures).

Sources & how we know this

Singapore-Cambridge GCE N(A)-Level Additional Mathematics (Syllabus 4051) — Singapore Examinations and Assessment Board (2026)