6.2 Laws of Logarithms

2026 Syllabus Objectives

By the end of these notes, you will be able to:

  • Know and use the laws of logarithms (the rules that let you simplify and manipulate logarithmic expressions)
  • Use the change of base formula to rewrite logarithms in a different base
  • Apply these skills to questions such as:
    • Writing 3 + 2 lg p − lg q as a single base-10 logarithm
    • Writing 1 / log₅ e as a natural logarithm

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1. Quick Recap — What Is a Logarithm?

Before we dive into the laws, make sure you remember what a logarithm actually means.

log_a(x) = y means the same thing as a^y = x

  • a is called the base (the number being raised to a power)
  • x is the argument (the number inside the log)
  • y is the answer — the power you raise the base to in order to get x

Example: log₂(8) = 3 because 2³ = 8

2. Special Notation You Must Know

Two bases are used so often that they have their own shorthand:

NotationMeaningCalled
lg xlog₁₀(x) — logarithm with base 10Common logarithm
ln xlogₑ(x) — logarithm with base e (≈ 2.718)Natural logarithm
  • lg is simply a shortcut for log₁₀. You will see this in exam questions.
  • ln uses the special number e (Euler's number, approximately 2.718). It appears everywhere in science and maths.
  • If no base is written, it is usually safe to assume base 10 — but always check the question.

3. Two Basic Results

These two results are true for any base a (where a > 0 and a ≠ 1):

Result 1: log_a(a) = 1

Why? Because a¹ = a. You raise the base to the power 1 to get the base itself.

Examples:

  • log₅(5) = 1
  • lg(10) = 1 (because 10¹ = 10)
  • ln(e) = 1 (because e¹ = e)

Result 2: log_a(1) = 0

Why? Because a⁰ = 1. Any number raised to the power 0 equals 1.

Examples:

  • log₃(1) = 0
  • lg(1) = 0
  • ln(1) = 0

4. The Three Laws of Logarithms

These are the most important rules. Learn them, understand what they mean, and practise using them.

Law 1 — The Multiplication Law (Addition of Logs)

log⁡a(xy)=log⁡a(x)+log⁡a(y)\log_a(xy) = \log_a(x) + \log_a(y)

In plain English: The log of a product (two things multiplied) equals the sum of their individual logs.

Example: lg⁡(100×5)=lg⁡(100)+lg⁡(5)=2+lg⁡(5)\lg(100 \times 5) = \lg(100) + \lg(5) = 2 + \lg(5)

Another example: log⁡2(4×8)=log⁡2(4)+log⁡2(8)=2+3=5\log_2(4 \times 8) = \log_2(4) + \log_2(8) = 2 + 3 = 5 Check: log₂(32) = 5 ✓ (because 2⁵ = 32)

Tip: This law also works in reverse — if you see two logs being added, you can combine them into one log by multiplying the arguments.

log⁡a(x)+log⁡a(y)=log⁡a(xy)\log_a(x) + \log_a(y) = \log_a(xy)

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