Year 10: Understanding logarithms

Logarithms can seem strange at first, especially because they look different from the types of numbers and operations you are used to. But once you understand the key ideas behind logarithms, they become powerful tools for solving equations, analysing patterns, and understanding real-world systems such as earthquakes, sound intensity, and population growth.

Here We will explore what logarithms are, why they exist, how they connect to indices, and how we use them for solving mathematical problems. By the end, you should have a strong understanding of the meaning of logarithms and be ready to apply them in your schoolwork.

1. What is a logarithm?

A logarithm answers a very specific question:

“To what power must I raise a base number to get a particular result?”

For example, consider the number 100. We know that:

$10 ² = 100$

So the logarithm of 100 base 10 is 2. We write this as:

$\log ₁ ₀ (100) = 2$

This means: “10 raised to the power of 2 gives 100.” More generally:

$a ˣ = b \leftrightarrow \log ₐ (b) = x$

A logarithm is simply the inverse of an exponential. If exponentials answer “What is the result when we raise a base to a power?”, then logarithms answer “What power gives this result?”

Examples:

$\log ₁ ₀ (1000) = 3$ because $10 ³ = 1000$
$\log ₂ (8) = 3$ because $2 ³ = 8$
$\log ₅ (125) = 3$ because $5 ³ = 125$

2. Understanding the base of a logarithm

Every logarithm has a base, which tells you which exponential relationship is being used. In school mathematics, the two most common bases are:

Base 10: $\log ₁ ₀ (x)$ also written as $\log (x)$ in many textbooks.
Base e: $\ln (x)$ , used in advanced mathematics involving continuous growth.

For Year 10, the focus is primarily on base 10 logarithms, although understanding the general definition will help you later.

Example: Find $\log ₂ (32)$ .

We want the number x such that $2 ˣ = 32$ .

Since $2 ⁵ = 32$ , we have $\log ₂ (32) = 5$ .

3. Why do we need logarithms?

Logarithms appear naturally in many real-life situations where growth is not linear but exponential. This means the quantity increases by multiplication rather than addition.

Some real-world examples include:

The Richter scale for measuring earthquake strength
pH levels in chemistry
Sound intensity in decibels
Population growth
Radioactive decay

Without logarithms, the numbers involved in these systems would be difficult to work with. Logarithms allow us to convert multiplicative relationships into additive ones, which simplifies calculations.

Example:

An earthquake of magnitude 6 is not “twice as strong” as a magnitude 3. It is thousands of times stronger because the scale is logarithmic.

4. Logarithms and indices: The inverse relationship

Logarithms and indices (powers) are inverse operations, just like multiplication and division.

If $a ˣ = b$ , then $\log ₐ (b) = x$ .

This relationship helps us move back and forth between exponential form and logarithmic form.

Example:

2 ⁴ = 16

can be rewritten as

\log ₂ (16) = 4

This means that logarithms allow us to solve exponential equations, especially when the unknown variable is in the exponent.

5. Evaluating logarithms

To evaluate a logarithm, you simply ask: “What exponent gives this number?”

5.1 Numbers that are powers of the base

$\log ₁ ₀ (1) = 0$ because $10 ⁰ = 1$
$\log ₁ ₀ (0.1) = - 1$ because $10 ⁻ ¹ = 0.1$
$\log ₂ (1 / 8) = - 3$ because $2 ⁻ ³ = 1 / 8$

Notice that logarithms can have negative results whenever the number inside is less than 1.

5.2 Evaluating by rewriting

Sometimes you must rewrite a number as a power to evaluate a logarithm.

Example:

Evaluate

\log ₃ (27)

Since $27 = 3 ³$ , we have $\log ₃ (27) = 3$ .

6. Laws of logarithms

Logarithms obey several important rules called the laws of logarithms. These rules are essential for simplifying expressions and solving equations.

6.1 Product Law

$\log ₐ (x y) = \log ₐ (x) + \log ₐ (y)$

This comes from the fact that multiplying inside the logarithm corresponds to adding exponents.

Example:

\log ₁ ₀ (100 \times 1000) = \log ₁ ₀ (100) + \log ₁ ₀ (1000)

= 2 + 3 = 5

6.2 Quotient Law

$\log ₐ (x / y) = \log ₐ (x) - \log ₐ (y)$

Example:

\log ₁ ₀ (1000 / 10) = \log ₁ ₀ (1000) - \log ₁ ₀ (10) = 3 - 1 = 2

6.3 Power Law

$\log ₐ (x ⁿ) = n \cdot \log ₐ (x)$

This rule is extremely useful for solving equations involving variables in exponents.

Example:

\log ₂ (8 ⁵) = 5 \cdot \log ₂ (8) = 5 \cdot 3 = 15

7. Solving logarithmic equations

7.1 Solving by converting to exponential form

Example:

Solve

\log ₄ (x) = 3

.

Convert to exponential form:

4 ³ = x

x = 64

7.2 Solving exponential equations using logs

Example:

Solve

2 ˣ = 40

.

Take log base 2 of both sides:

x = \log ₂ (40)

.

Evaluate using change of base:

x = \log ₁ ₀ (40) / \log ₁ ₀ (2)

8. Change of base formula

Calculators often only have log (base 10) and ln buttons. The change of base formula lets you compute logarithms of any base.

$\log ₐ (b) = \log ₁ ₀ (b) / \log ₁ ₀ (a)$

Example:

Find

\log ₂ (50)

on a calculator:

\log ₂ (50) = \log (50) / \log (2)

9. Logarithmic scales in real life

9.1 Earthquakes: Richter scale

The Richter scale is logarithmic: a magnitude 6 earthquake is ten times stronger than a magnitude 5.

9.2 Sound intensity: decibels

The decibel (dB) scale also uses logarithms. A small increase in decibels represents a large increase in loudness.

9.3 Acidity: pH scale

The pH scale in chemistry measures acidity using a base-10 logarithm:

$p H = - \log ₁ ₀ ([H ⁺])$

10. Common mistakes

Thinking log(x) means log × x:

Logarithms are functions, not multipliers. $\log (5)$ means “the log of 5”, not “log times 5”.

Forgetting the base:

Always be clear which base is being used. In Year 10 maths, unless stated otherwise, assume base 10.

Trying to take logs of negative numbers:

You cannot take logarithms of negative numbers or zero in the real number system. The expression inside a logarithm must be positive.

Messing up the laws of logs:

A very common mistake is attempting to “break” logarithms incorrectly:

❌ $\log (x + y) = \log (x) + \log (y)$ (This is NOT true!)
❌ $\log (x - y) = \log (x) - \log (y)$ (Also false.)

Only multiplication, division, and powers work with log laws.

11. Summary

Logarithms are the inverse of exponentiation.
logₐ(b) asks: “To what power must I raise a to get b?”
Base 10 logs are most common in Year 10.
Negative logarithms occur when the number inside is less than 1.
The laws of logs help simplify expressions:
- $\log (x y) = \log (x) + \log (y)$
- $\log (x / y) = \log (x) - \log (y)$
- $\log (x ⁿ) = n \log (x)$
Real-world applications include earthquakes, sound, pH, finance, and population models.

Learning logarithms now will give you a strong foundation for advanced mathematics, including algebra, calculus, physics, chemistry, biology, and financial maths. Logarithms are one of the most important mathematical tools in science and technology, and understanding them opens the door to many exciting areas of study.