---
title: "A-Maths Indices, Surds, Logarithms & Exponentials"
description: "Indices, surds and logarithms are the algebra engine of A-Maths. This guide covers the laws, change of base, rationalising surds, and solving exponential and logarithmic equations."
author: "Gabriel"
author_url: "https://ancourage.academy/authors/gabriel"
published_at: 2026-07-13
modified_at: 2026-07-13
category: "teaching"
tags: ["Mathematics", "Secondary", "O-Level", "A-Maths", "SEC", "Logarithms", "Indices", "Singapore", "Exam Tips"]
canonical: "https://ancourage.academy/articles/a-maths-indices-surds-logarithms-exponentials-singapore"
source: "https://ancourage.academy/articles/a-maths-indices-surds-logarithms-exponentials-singapore"
language: "en-SG"
word_count: 1582
reading_time: "PT8M"
cover_image: "https://ancourage.academy/academic-pic/IMG_0140.jpg"
reviewed_by: "Min Hui"
---

# A-Maths Indices, Surds, Logarithms & Exponentials

Indices, surds and logarithms are the algebra engine of A-Maths. This guide covers the laws, change of base, rationalising surds, and solving exponential and logarithmic equations.

**Indices, surds and logarithms are the algebra engine of A-Maths — once the three sets of laws are secure, the equations that look intimidating become routine.** The key insight students miss is that logarithms and indices are two ways of writing the same relationship, so a problem that is hard in one form is often easy in the other. This guide is from [Ancourage Academy](https://ancourage.academy/academy), whose [secondary A-Maths tuition](https://ancourage.academy/courses/academy/secondary/a-maths) teaches these laws method-first in small groups of 3–6 at [Bishan](https://ancourage.academy/find-us/bishan) and [Woodlands](https://ancourage.academy/find-us/woodlands).

This is a single-topic deep-dive — a sibling to our [A-Maths binomial theorem](https://ancourage.academy/articles/a-maths-binomial-theorem-expansion-guide-singapore) and [quadratics](https://ancourage.academy/articles/a-maths-quadratic-functions-equations-inequalities-singapore) guides. If you are still deciding whether to take A-Maths, read [E-Maths vs A-Maths](https://ancourage.academy/articles/e-maths-vs-a-maths-difference-singapore) first.

**If logarithms are where your child's A-Maths confidence broke, Ancourage Academy's [Sec 3 A-Maths programme](https://ancourage.academy/courses/academy/secondary/s3/a-maths) rebuilds the laws from first principles — [book a trial class (usually $18)](https://ancourage.academy/trial-class) for a diagnostic assessment.**

## What Does This Topic Cover in A-Maths?

**In O-Level / SEC A-Maths, this strand brings together surds — including the four operations, rationalising the denominator and solving equations involving surds — logarithms including change of base, the equivalence of exponential and logarithmic form, and the graphs of exponential and logarithmic functions, all resting on the laws of indices assumed from O-Level Mathematics.** The [SEAB A-Maths syllabus (4049)](https://www.seab.gov.sg/gce-o-level/o-level-syllabuses-examined-for-school-candidates-2026/) defines exactly what is examinable. From 2027 the same content carries over to the SEC G3 A-Maths syllabus (K341) unchanged.

## How Do You Work With Surds?

**A surd is an irrational root left in exact form, such as √2, and the central skill is rationalising the denominator — removing a surd from the bottom of a fraction by multiplying by a suitable form of 1.**

-   **Single surd denominator:** multiply top and bottom by that surd, e.g. 1/√3 = √3/3.
-   **Binomial surd denominator:** multiply by the conjugate, e.g. for 1/(2 + √3), multiply by (2 − √3)/(2 − √3) so the denominator becomes the difference of two squares.

The conjugate trick works because (a + √b)(a − √b) = a² − b, which is rational. Keep surds in exact form throughout — converting to decimals early loses accuracy and, in "show that" questions, loses marks.

## What Are the Laws of Indices?

**The laws of indices govern how powers combine, and every index manipulation in A-Maths is one of these rules applied carefully.**

| Law | Rule |
| --- | --- |
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ |
| Zero index | a⁰ = 1 (a ≠ 0) |
| Negative index | a⁻ⁿ = 1 / aⁿ |
| Fractional index | a^(m/n) = ⁿ√(aᵐ) |

To solve an exponential equation where the unknown is in the power, first try to write both sides with the same base — for example, 2^(x+1) = 8 becomes 2^(x+1) = 2³, so x + 1 = 3. When the bases cannot be matched, you take logarithms instead.

## What Are the Laws of Logarithms?

**A logarithm answers the question "to what power must the base be raised?" — logₐ y = x means exactly aˣ = y — and the laws of logarithms mirror the laws of indices.**

| Law | Rule |
| --- | --- |
| Product | logₐ(xy) = logₐ x + logₐ y |
| Quotient | logₐ(x/y) = logₐ x − logₐ y |
| Power | logₐ(xⁿ) = n logₐ x |
| Change of base | logₐ b = (log\_c b) / (log\_c a) |
| Special values | logₐ 1 = 0 and logₐ a = 1 |

Change of base is the law students forget most often, yet it is essential whenever an equation mixes logarithms of different bases — convert them all to a common base first. The natural logarithm ln x (base e) and its inverse eˣ obey exactly the same laws, and eˣ appears throughout exponential-growth and decay questions.

## How Do You Solve Logarithmic and Exponential Equations?

**The strategy depends on where the unknown sits: take logarithms when the unknown is in the exponent, and exponentiate (undo the log) when the unknown is inside a logarithm.**

1.  **Unknown in the power (e.g., 5ˣ = 12):** take logarithms of both sides, then x = (log 12)/(log 5).
2.  **Unknown inside a log (e.g., log₂(x − 1) = 3):** rewrite in index form, so x − 1 = 2³ = 8, giving x = 9.
3.  **Combine first:** use the laws to condense several log terms into a single logarithm before solving.
4.  **Check the domain:** a logarithm is only defined for a positive argument — always reject solutions that make any log negative or zero.

## What Do Exponential and Logarithmic Graphs Look Like?

**The graph of y = aˣ (a > 1) rises steeply and passes through (0, 1); the graph of y = logₐ x is its mirror image in the line y = x, passing through (1, 0).** Knowing the shape, key points and asymptotes lets you sketch quickly and check whether an algebraic answer is reasonable. The exponential graph has the x-axis as a horizontal asymptote; the logarithmic graph has the y-axis as a vertical asymptote and is only defined for x > 0.

## The Most Common Indices and Logarithms Mistakes

**In our A-Maths classes at Ancourage Academy, a handful of recurring errors cause most avoidable mark loss in this topic.**

| Mistake | Why it happens | How to fix it |
| --- | --- | --- |
| log(x + y) = log x + log y | Misremembering the product law | The product law applies to log(xy), never to log(x + y) |
| Forgetting change of base | Trying to combine logs of different bases directly | Convert all logs to one base before applying the other laws |
| Keeping invalid solutions | Not checking the log domain | Reject any solution that makes a logarithm's argument ≤ 0 |
| Rationalising errors | Multiplying by the wrong conjugate | Use (a − √b) for (a + √b); the denominator becomes a² − b |
| Mishandling negative/fractional indices | Reading a⁻ⁿ as −aⁿ | a⁻ⁿ = 1/aⁿ; a^(m/n) is the n-th root of aᵐ |

## How Does This Topic Connect to the Rest of A-Maths?

**Indices and logarithms thread through several A-Maths topics and are assumed knowledge in JC.**

-   **Linear law:** taking logarithms is exactly how non-linear relationships such as y = axⁿ are turned into straight lines. See our [A-Maths linear law guide](https://ancourage.academy/articles/a-maths-linear-law-singapore).
-   **Calculus:** differentiating eˣ and ln x, and integrating eˣ, is a core part of A-Maths calculus. See our [calculus deep-dive](https://ancourage.academy/articles/a-maths-calculus-differentiation-integration-singapore).
-   **Foundation for JC:** the same laws power exponential models and logarithmic differentiation in [H2 Mathematics](https://ancourage.academy/articles/h2-mathematics-jc-guide-singapore).

## A Study Plan for Mastering Indices, Surds and Logarithms

**Master this topic in order: surds and index laws first, then logarithm laws, then equation-solving, then graphs.**

1.  **Week 1 — surds and indices:** drill rationalising denominators and all six index laws until recall is instant.
2.  **Week 2 — logarithm laws:** practise the product, quotient, power and change-of-base laws, and the index–log equivalence.
3.  **Week 3 — equations:** solve equations involving surds (checking for extraneous roots) and exponential and logarithmic equations (checking the domain).
4.  **Week 4 — graphs and mixed practice:** sketch exponential and log graphs and tackle combined questions under timed conditions.

Ancourage Academy's [Sec 3](https://ancourage.academy/courses/academy/secondary/s3/a-maths) and [Sec 4 A-Maths](https://ancourage.academy/courses/academy/secondary/s4/a-maths) programmes work through this topic on exactly this progression in small groups of 3–6. If logarithms are where your child got stuck, our [A-Maths survival guide](https://ancourage.academy/articles/a-maths-survival-guide-struggling-additional-maths-singapore) covers the wider recovery plan — book a [trial class (usually $18)](https://ancourage.academy/trial-class) for a diagnostic, or [WhatsApp us](https://api.whatsapp.com/send/?phone=6588498106&type=phone_number&app_absent=0) with any questions.

## Common Questions About A-Maths Indices, Surds and Logarithms

### What is the change of base rule and when do you use it?

The change of base rule is logₐ b = (log\_c b)/(log\_c a), which rewrites a logarithm in terms of any other base c. You use it whenever an equation contains logarithms of different bases — convert them all to a single common base, then apply the product, quotient and power laws. Forgetting change of base is the most common reason students get stuck on mixed-base log equations.

### How do you solve an equation with the unknown in the power?

First try to express both sides with the same base; for example 3^(2x) = 27 becomes 3^(2x) = 3³, so 2x = 3. If the bases cannot be matched, take logarithms of both sides and rearrange: for 5ˣ = 12, x = (log 12)/(log 5). Either approach turns an exponential equation into a linear or quadratic one you already know how to solve.

### How do you rationalise a denominator with a surd?

For a single surd such as 1/√3, multiply numerator and denominator by that surd to get √3/3. For a binomial denominator such as 1/(2 + √3), multiply by the conjugate (2 − √3)/(2 − √3); the denominator becomes 2² − 3 = 1, which is rational. The conjugate works because (a + √b)(a − √b) = a² − b.

### Why must you check the domain of a logarithmic equation?

A logarithm is only defined for a positive argument, so logₐ(negative) and logₐ(0) do not exist. When solving, you may produce candidate values that make a log's argument zero or negative — these are extraneous and must be rejected. Always substitute your answers back to confirm every logarithm in the original equation is defined.

### Is this topic the same under SEC from 2027?

Yes. Surds, indices and logarithms move from the O-Level A-Maths syllabus (4049) to the SEC G3 A-Maths syllabus (K341) from 2027 with no change to the laws, change of base, or equation-solving requirements. The "O-Level / SEC" dual reference reflects this transition.

Related: [A-Maths Linear Law](https://ancourage.academy/articles/a-maths-linear-law-singapore) · [A-Maths Binomial Theorem](https://ancourage.academy/articles/a-maths-binomial-theorem-expansion-guide-singapore) · [A-Maths Calculus Guide](https://ancourage.academy/articles/a-maths-calculus-differentiation-integration-singapore) · [A-Maths Survival Guide](https://ancourage.academy/articles/a-maths-survival-guide-struggling-additional-maths-singapore) · [Choosing between E-Maths and A-Maths](https://ancourage.academy/articles/e-maths-vs-a-maths-difference-singapore) · [A-Maths Polynomials & Partial Fractions guide](https://ancourage.academy/articles/a-maths-polynomials-partial-fractions-singapore)

## Related Courses

- [Sec 3 O-Level / SEC A-Maths](https://ancourage.academy/courses/academy/secondary/s3/a-maths) — Laws of indices, surds and logarithms in small groups of 3–6
- [Sec 4 O-Level / SEC A-Maths](https://ancourage.academy/courses/academy/secondary/s4/a-maths) — Exponential and logarithmic equations and exam preparation
- [Secondary A-Maths Programme](https://ancourage.academy/courses/academy/secondary/a-maths) — All A-Maths courses by level at Bishan and Woodlands
- [Trial Class (Usually $18)](https://ancourage.academy/trial-class) — Diagnostic assessment of your child’s logarithm foundations

## Sources

- [O-Level Additional Mathematics Syllabus 4049 (seab.gov.sg)](https://www.seab.gov.sg/gce-o-level/o-level-syllabuses-examined-for-school-candidates-2026/) — Singapore Examinations and Assessment Board
- [SEC G3 Additional Mathematics Syllabus K341 (seab.gov.sg)](https://www.seab.gov.sg/secondary-education-certificate-sec/g3-syllabuses-for-school-candidates-2027/) — Singapore Examinations and Assessment Board
- [Secondary Mathematics Syllabuses (moe.gov.sg)](https://www.moe.gov.sg/secondary/courses) — Ministry of Education, Singapore