Inverting the Power: Mastering Logarithmic Laws

By Joshua Ramirez, MSc·Updated April 18, 2026
A blackboard covered in logarithmic equations and exponential graphs.

What are the 3 laws of logarithms I must memorize?

Law 1 (Product Rule): log(A x B) = log A + log B. Law 2 (Quotient Rule): log(A / B) = log A - log B. Law 3 (Power Rule): log(A^n) = n x log A. Additionally, always remember: log(1) = 0, and log base a of a = 1. These 5 facts handle every single logarithm question in the CAIE exam.

Table of Contents

Logarithms terrify students because they look alien. But a log is just a backwards exponent. If you understand that 2^5 = 32, then you already understand that log base 2 of 32 = 5. This guide from our Ultimate Add Maths Guide strips away the fear.

1. What is a Logarithm? (The Inverse)

An exponent answers: "What do I get when I raise 2 to the power of 5?" Answer: 32. A logarithm asks the reverse: "What power must I raise 2 to, in order to get 32?" Answer: 5.

The Conversion: a^x = b ↔ log_a(b) = x

2^3 = 8 ↔ log_2(8) = 3

10^2 = 100 ↔ log_10(100) = 2

💡 Tutor's Tip
The "Base-Answer-Power" Mnemonic: Read log_2(8) = 3 as: "The Base is 2. The Answer is 8. The Power is 3." Then reconstruct: Base^Power = Answer → 2^3 = 8. Use this to convert in either direction instantly.

2. The 3 Core Log Laws

Law 1: Product Rule

log(A × B) = log A + log B. Multiplication inside becomes addition outside.

Law 2: Quotient Rule

log(A / B) = log A - log B. Division inside becomes subtraction outside.

Law 3: Power Rule

log(A^n) = n × log A. The power slides down to become a multiplier.

💡 Tutor's Tip
Common Trap: log(A + B) does NOT equal log A + log B. The laws only work for multiplication, division, and powers inside the log. Addition and subtraction inside the log cannot be separated. This mistake costs students 2-3 marks every single paper.

3. Solving Exponential Equations

When the unknown is in the exponent (like 5^x = 200), you cannot solve it algebraically. You must use logarithms to "bring the power down".

Worked Example: Solve 3^(2x+1) = 50

Step 1: Take log of both sides: log(3^(2x+1)) = log(50)

Step 2: Apply Power Rule: (2x+1) × log(3) = log(50)

Step 3: Isolate: 2x+1 = log(50)/log(3) = 3.561

Step 4: Solve: x = (3.561 - 1)/2 = 1.28

4. The Change of Base Formula

Your calculator only does log base 10 (the "log" button) and log base e (the "ln" button). If the exam gives you log base 5 of 12, you must convert it.

Formula: log_a(b) = log(b) / log(a)

log_5(12) = log(12) / log(5) = 1.079 / 0.699 = 1.544

Joshua Ramirez📋 From the Desk of Joshua Ramirez
The Natural Log (ln):ln is just log base e, where e ≈ 2.718. The CAIE exam uses it in growth/decay questions. If you see e^(0.5t) = 100, take ln of both sides: 0.5t = ln(100). Your calculator has a dedicated "ln" button — use it instead of trying change of base.

Frequently Asked Questions

What is a Logarithm?▼
The inverse of an exponent. It answers: 'What power must I raise this base to, in order to get this number?'
What are the 3 core Log Laws?▼
Product: log(AB) = logA + logB. Quotient: log(A/B) = logA - logB. Power: log(A^n) = n*logA.
How do I solve exponential equations using logs?▼
Take log of both sides, apply the Power Rule to bring the exponent down, then solve the resulting linear equation algebraically.
What is the Change of Base formula?▼
log_a(b) = log(b)/log(a). Converts any base to base 10 so your calculator can evaluate it.

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