Singapore Math guide to exponential and logarithmic functions, an upper secondary Additional Mathematics topic, for families and teachers outside Singapore.
A logarithm answers one question: what power?
Once a student hears that question inside every log, the laws stop looking like rules to memorise.
In Singapore Mathematics, exponential and logarithmic functions are taught in G3 Additional Mathematics, an elective taken in Secondary 3 to 4. Students work with ax, ex, loga x and ln x, learn the laws of logarithms and change of base, and solve equations and modelling problems.
The same functions return in Linear Law and in the calculus topics that close the course.
This guide is useful for families and teachers who want to:
- understand what the Additional Mathematics syllabus expects;
- teach logarithms as the inverse of powers, not as a list of tricks;
- prepare a student who is moving into a Singapore secondary school; or
- support a student using Singapore Math materials in another country.
Quick facts
Level: Secondary 3 to 4 (G3 Additional Mathematics, an elective)
Strand: Algebra
Official source: SEAB G3 Additional Mathematics, Syllabus K341 (SEC, from 2027). The 2026 O-Level Additional Mathematics syllabus 4049 has the same content.
Comes after: Indices and Binomial Theorem
Leads to: Linear Law and Differentiation
What Exponential and Logarithmic Functions Mean in Singapore Mathematics
An exponential function has the variable in the power, as in y = 2x or y = ex.
A logarithm reverses that. If 23 = 8, then log2 8 = 3. The log tells you the power that the base must be raised to.
The syllabus states this link directly as the equivalence of y = ax and x = loga y. Students move between the two forms constantly.
Two bases matter most. Base 10 is the “log” key on a calculator. Base e, a constant close to 2.718, gives the natural logarithm ln x.
The number e matters because ex and ln x behave very simply under differentiation, which students meet later in the same course.
Where Exponential and Logarithmic Functions Sit in the Singapore Syllabus
Additional Mathematics has three strands: Algebra, Geometry and Trigonometry, and Calculus. Exponential and logarithmic functions belong to the Algebra strand.
In G3 Additional Mathematics, students learn:
- the functions ax, ex, loga x and ln x, and their graphs;
- the laws of logarithms;
- the equivalence of y = ax and x = loga y;
- change of base of logarithms;
- simplifying expressions and solving simple equations involving these functions; and
- using exponential and logarithmic functions as models.
| Course | What students do |
|---|---|
| G3 Mathematics, Sec 3 or 4 | Zero, negative and fractional indices; laws of indices; graphs of y = kax, where a is a positive integer |
| G3 Additional Mathematics, Sec 3 to 4 | ax, ex, loga x and ln x; laws of logarithms; change of base; equations; models |
| Additional Mathematics calculus | Derivatives of ex and ln x; integrals involving ex and eax + b |
MOE describes G2 Additional Mathematics as covering linear, quadratic and trigonometric functions, with G3 adding exponential and logarithmic functions. This page describes G3.
How Singapore Teaches Exponential and Logarithmic Functions
1. Start from powers
Students list powers they already know: 21 = 2, 22 = 4, 23 = 8. Then they read each fact backwards as a logarithm. This links straight to Indices.
2. Draw the two graphs together
y = 2x passes through (0, 1) and approaches the x-axis. y = log2 x passes through (1, 0) and approaches the y-axis. Each is the reflection of the other in the line y = x.
3. Derive the laws from index laws
loga (xy) = loga x + loga y comes from multiplying powers by adding indices. The quotient and power laws follow the same way: loga (x/y) = loga x − loga y and loga xn = n loga x.
4. Change base when the calculator needs it
loga b = (log b)/(log a), using any base the calculator offers. Students also use it to rewrite one base in terms of another.
5. Solve equations by choosing a form
Some equations need logs taken of both sides. Others combine logs into a single log first, then switch to index form.
6. Model growth and decay
Students use forms such as N = N0ekt for growth and decay, then interpret the answer in context.
Worked Examples
Solving a logarithmic equation
Question: Solve log2 (x + 3) + log2 (x − 1) = 5.
Step 1. Combine into one log: log2 [(x + 3)(x − 1)] = 5.
Step 2. Switch to index form: (x + 3)(x − 1) = 25 = 32.
Step 3. Expand: x² + 2x − 3 = 32, so x² + 2x − 35 = 0.
Step 4. Factorise: (x + 7)(x − 5) = 0, so x = 5 or x = −7.
Step 5. Reject x = −7, because it makes x − 1 negative and the log of a negative number is undefined.
Check: When x = 5, log2 8 + log2 4 = 3 + 2 = 5.
An exponential model
Question: The value of a machine after t years is V = 50 000e−0.2t dollars. After how many years does its value first fall to $20 000?
Step 1. Set up: 50 000e−0.2t = 20 000, so e−0.2t = 0.4.
Step 2. Take natural logs: −0.2t = ln 0.4.
Step 3. t = (ln 0.4) ÷ (−0.2) = 4.58 years, correct to 3 significant figures.
Check: e−0.2 × 4.58 is about 0.400, and 50 000 × 0.400 = 20 000.
In context, a student should also say what the answer means: the value reaches $20 000 during the fifth year.
Common Mistakes
- Splitting the log of a sum. log (a + b) is not log a + log b. Fix: the addition law only works for a product, log (ab).
- Keeping impossible roots. A solution that makes any log argument zero or negative must be rejected. Fix: substitute every root into the original equation.
- Confusing (log x)² with log x². The power law gives log x² = 2 log x, but (log x)² is a different expression. Fix: write brackets clearly.
- Turning change of base upside down. log3 20 is (log 20)/(log 3), not (log 3)/(log 20). Fix: the base of the old log goes underneath.
- Drawing ln x across the y-axis. The graph of y = ln x exists only for x > 0. Fix: mark the asymptote x = 0 and the point (1, 0) first.
- Rounding too early. Rounding ln 0.4 to −0.9 changes the final answer. Fix: keep full calculator values until the last step.
Teaching Exponential and Logarithmic Functions Outside Singapore
- The subject name. Additional Mathematics is a separate elective taken alongside Mathematics in upper secondary. Singapore students often call it A-Math.
- Notation. Singapore materials write loga x for a general base and ln x for base e. Calculators usually label base 10 as “log”.
- Timing. Logarithms here sit in the upper secondary elective. Your local course may place them differently, so decide by readiness. The Singapore Math Grade Levels guide explains why direct conversions are unreliable.
- Prerequisites. Students need secure laws of indices, including fractional and negative indices, and confident factorising of quadratics.
- Calculators. An approved calculator may be used in both Additional Mathematics papers. Students still need exact work, such as log2 32 = 5, and must show essential working.
How to Practise Exponential and Logarithmic Functions at Home
- Read five power facts aloud as logs: “2 to the 5 is 32, so log base 2 of 32 is 5.”
- Sketch y = 2x and y = log2 x on one set of axes and mark (0, 1), (1, 0) and the line y = x.
- Simplify two expressions using the laws, saying which law each step uses.
- Solve one log equation and check every root in the original.
- Solve one equation such as 3x = 20 by taking logs, then check by substitution.
- Once a week, work one growth or decay model and write one sentence interpreting the answer.
Twenty focused minutes is enough for one session.
Try These
- Find the exact value of log3 81.
- Write 2 log 5 + log 4 as a single logarithm and evaluate it, where log means base 10.
- Solve 5x = 40, giving x correct to 3 significant figures.
- Given that log2 3 = p, express log8 9 in terms of p.
Answers
- 4, because 34 = 81.
- log 100 = 2. First 2 log 5 = log 25, then log 25 + log 4 = log 100.
- x = 2.29, because x = (ln 40)/(ln 5) = 2.292…
- 2p/3. Change base to 2: log8 9 = (log2 9)/(log2 8) = 2p/3.
What Comes Next
Exponential and logarithmic functions are used again within the same course.
- Linear Law uses logs to turn y = axn and y = kbx into straight lines.
- Differentiation includes the derivatives of ex and ln x.
- Integration includes integrals of ex and eax + b.
A student who reads every log as “what power?” carries a clear picture into each of these.
Frequently Asked Questions
When do Singapore students learn logarithms?
In G3 Additional Mathematics, the upper secondary elective taken in Secondary 3 to 4. G3 Mathematics prepares the ground with indices and graphs of y = kax.
Is ex part of the syllabus?
Yes. The syllabus lists ax, ex, loga x and ln x, with their graphs.
What does “the equivalence of y = ax and x = loga y” mean?
The two statements say the same thing. Students switch between them to solve equations and to sketch graphs.
Does G2 Additional Mathematics include logarithms?
MOE describes G3 Additional Mathematics as adding exponential and logarithmic functions to the linear, quadratic and trigonometric functions in G2. Check the current G2 syllabus if your child takes G2.
Can students use a calculator?
Yes. The examination has two papers of 2 hours 15 minutes, each worth 90 marks, and an approved calculator may be used in both. A formula list is provided, but the laws of logarithms are not on it.
Why does my child get the right method but the wrong answer?
Usually a root was not checked, or a value was rounded too early. Build both checks into every question.
Helpful Reading
- Singapore Mathematics | International Edition: Indices
- Singapore Mathematics | International Edition: Cubic, Reciprocal and Exponential Graphs
- Singapore Mathematics | International Edition: Linear Law
- Why Logarithms Can Feel Like Mathematics Written Backwards
- Secondary 3 Additional Mathematics | Exponential and Logarithmic Functions
- Additional Mathematics Synthesis Guide 33: Exponential and Logarithmic Functions
- How to Make Sense of the SEC G3 Additional Mathematics Syllabus
- SEAB G3 Additional Mathematics Syllabus K341
Exponential and Logarithmic Functions in Singapore Mathematics
A power and a log are two views of one fact.
Read every log as “what power?”
Combine, then switch forms.
Check every root.
Round only at the end.
Taught this way, logarithms stop feeling backwards.
They become the tool that opens Linear Law and calculus.
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Singapore Mathematics | International Edition is a Bukit Timah Tutor guide series for families and teachers outside Singapore. It is not a textbook and is not affiliated with MOE, SEAB or any publisher. Syllabus details were checked against MOE and SEAB documents in October 2026. Always confirm current requirements with your school.
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