IP AMaths Notes (Upper Sec, Year 3-4): 02) Logarithms and Exponentials

Study guide08 Nov 2025, 00:00 ZUpdated 17 Jul 2026

Log laws, change of base, and exponential modelling with inverse relationships for IP Additional Mathematics.

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For Integrated Programme students: Your current school materials, teacher instructions, and assessment scope take precedence because IP topic sequence and depth vary by school. This is an Eclat IP guide, not the O-Level / SEC G3 exam-track guide.

How this chapter applies

Eclat core: exponential and logarithmic functions, graphs, laws, change of base, equations, and models form the main route.
School-sensitive extension: more demanding parameter models, asymptotic analysis, and links to calculus may vary by school.
2027 national comparison: K341 Topic A6 covers exponential and logarithmic functions, their graphs, logarithm laws, change of base, simple equations, and models.
Check your school: confirm the required bases, graph sketching conventions, and modelling depth.
Exam-track route: use the separate O-Level and SEC G3 Additional Mathematics notes for K341 topic ownership.

Q: What does IP AMaths Notes (Upper Sec, Year 3-4): 02) Logarithms and Exponentials cover?
A: Log laws, change of base, and exponential modelling with inverse relationships for IP Additional Mathematics.

Logarithms invert exponentials. Switch comfortably between both forms to linearise growth models and solve equations involving powers.

Keep the full topic roadmap handy via our IP Maths tuition hub so you can jump into related drills, quizzes, or diagnostics as you move through these notes.

New to the Integrated Programme? Start with What is IP? | Browse all free IP notes.

The core idea is simple: Logarithms undo exponentials.

Use it as a working check: Before using log laws, check the domain: every logged expression must be positive. After solving, reject any root that breaks that condition.

Then go one layer deeper: Example: if log(x + 3) + log(x - 1) appears, the domain is x greater than 1 before any quadratic solving starts.

Choosing the first transformation

Logarithm questions usually become short once you choose the right form. Start by identifying what needs to be isolated or straightened.

Question cue

First transformation

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Marcus Pang·Managing Director (Maths)

Original form	First domain condition	Why this comes first
$\log_a(x - 2)$	$x > 2$	The expression inside the log must be positive before any solving starts.
$\log_a(x + 3) + \log_a(x - 1)$	$x > 1$	Both original log inputs must be positive, not just the combined product.
$\log_a((x - 4)^2)$	$x \neq 4$
$\log_a!\left(\dfrac{x + 2}{x - 5}\right)$	$\dfrac{x + 2}{x - 5} > 0$

IP AMaths Notes (Upper Sec, Year 3-4): 02) Logarithms and Exponentials

How this chapter applies

Choosing the first transformation

Sources

Domain Checkpoint Before Log Laws

1 Core facts

2 Worked example - Solve logarithmic equation

3 Worked example - Linear law

Linear-law axis checkpoint

4 Worked example - Exponential growth time

5 Practice Quiz

6 Try this

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