Lesson 1: Logarithms and Their Rules

👨🏽‍🏫 Introduction

Ever wondered how scientists and engineers deal with huge numbers like 1,000,000,000 or tiny ones like 0.00000001? Or why your calculator has a button that says log? 🤔

Welcome to the world of Logarithms — the smart shortcut to working with powers! Just like multiplication simplifies repeated addition, logarithms simplify powers and exponents. They’re especially useful when dealing with population growth, earthquakes, sound intensity, or compound interest! 🌍🔊📈

In this lesson, you’ll learn to:

Understand what a logarithm is
Convert between exponential and logarithmic forms
Apply the three main laws of logarithms
Solve logarithmic equations

🧠 Key Concepts

📌 1. What Is a Logarithm?

A logarithm is the power you raise a number (called the base) to in order to get another number.

Definition:

If $png$ , then $png$

a is the base (must be > 0 and ≠ 1)
b is the number (must be > 0)
x is the exponent

Example:
$png$ because $png$

🔁 2. Converting Between Forms

Logarithms and exponents are just two sides of the same coin! One “asks for the power”, the other “gives the result”.

Exponential to Logarithmic:

Suppose you know: $png$
You can rewrite it as a logarithm like this:
$png$

✅ Here’s what it means: “The power you raise 5 to, to get 25, is 2.”

Logarithmic to Exponential:

Suppose you know: $png$
This means 3 raised to the power of 4 gives 81. So in exponential form:
$png$

🔑 Always remember:
If $png$ , then $png$

📐 3. The 3 Laws of Logarithms

These rules make working with logarithms easier — especially when simplifying or solving expressions involving multiplication, division, or powers.

✅ Law 1: Product Rule

$png$

Why it works: Multiplying two numbers with the same base is like adding exponents. Logs reverse that — so multiplication turns into addition.

Example:
$png$

✅ Law 2: Quotient Rule

$png$

Why it works: Dividing powers of the same base subtracts the exponents. Since logs are based on powers, division becomes subtraction.

Example:
$png$

✅ Law 3: Power Rule

$png$

Why it works: Powers can be “brought down” in logs because of how exponents work. This makes solving easier!

Example:
$png$

🧮 4. Common and Natural Logs

Some logarithms are used so frequently that they have special names and keys on calculators:

Common Log: This means log base 10. It’s written simply as $png$ .
For example:
$png$ because $png$
Natural Log: This uses the base $e \approx 2.718$ . It’s written as $png$ .
Used mostly in higher science, growth models, and calculus.

🎯 For WASSCE: Focus mainly on base 10 (common logs) and small whole-number bases like 2, 3, 4, or 5. You’ll mostly use them to simplify, expand, or solve equations.

🧪 Worked Examples

Evaluate: $png$
Solution: What power of 2 gives 32?
$png$
Answer: $png$
Simplify: $png$
Apply Product Rule:
$png$
Expand: $png$
Apply Quotient Rule:
$png$
Solve: $png$
Convert to exponential:
$png$
Simplify: $png$
Apply Power Rule:
$png$