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, whose secondary A-Maths tuition teaches these laws method-first in small groups of 3–6 at Bishan and Woodlands.
This is a single-topic deep-dive — a sibling to our A-Maths binomial theorem and quadratics guides. If you are still deciding whether to take A-Maths, read E-Maths vs A-Maths first.
If logarithms are where your child's A-Maths confidence broke, Ancourage Academy's Sec 3 A-Maths programme rebuilds the laws from first principles — book a trial class (usually $18) 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) 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.
- Unknown in the power (e.g., 5ˣ = 12): take logarithms of both sides, then x = (log 12)/(log 5).
- Unknown inside a log (e.g., log₂(x − 1) = 3): rewrite in index form, so x − 1 = 2³ = 8, giving x = 9.
- Combine first: use the laws to condense several log terms into a single logarithm before solving.
- 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.
- Calculus: differentiating eˣ and ln x, and integrating eˣ, is a core part of A-Maths calculus. See our calculus deep-dive.
- Foundation for JC: the same laws power exponential models and logarithmic differentiation in H2 Mathematics.
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.
- Week 1 — surds and indices: drill rationalising denominators and all six index laws until recall is instant.
- Week 2 — logarithm laws: practise the product, quotient, power and change-of-base laws, and the index–log equivalence.
- Week 3 — equations: solve equations involving surds (checking for extraneous roots) and exponential and logarithmic equations (checking the domain).
- Week 4 — graphs and mixed practice: sketch exponential and log graphs and tackle combined questions under timed conditions.
Ancourage Academy's Sec 3 and Sec 4 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 covers the wider recovery plan — book a trial class (usually $18) for a diagnostic, or WhatsApp us 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 · A-Maths Binomial Theorem · A-Maths Calculus Guide · A-Maths Survival Guide · Choosing between E-Maths and A-Maths · A-Maths Polynomials & Partial Fractions guide