Logarithms: meaning and the link with indices

If $2^5 = 32$, then we say "the logarithm of $32$ to base $2$ is $5$." Logarithms are the inverse of indices — they answer the question: "to what power must I raise this base to get this number?" If indices are about computing $2^5$, logarithms are about asking "$2$ to what power gives $32$?"

Logarithms were invented in the early $1600$s by a Scottish mathematician, John Napier, to make arithmetic easier. They turn multiplication into addition and division into subtraction — a huge advantage before calculators existed. Even today, with calculators everywhere, WAEC still tests logarithms heavily because the underlying ideas appear in algebra, calculus, exponential growth, and many real-world applications.

This lesson defines logarithms, shows their connection to indices, introduces logarithm notation, and works through fundamental conversions.

Learning objectives

By the end of this lesson you should be able to:

• explain what a logarithm is
• convert between index form and logarithm form
• evaluate simple logarithms without a calculator
• recognise common logarithms (base 10)

The relationship between indices and logarithms

Logarithms and indices say the same thing in different ways:

Whenever you see a logarithm, you can rewrite it as an exponent. Whenever you see an exponent equation, you can rewrite it as a logarithm. The two forms are interchangeable.

Figure 1. The logarithm undoes the exponential: the two graphs are mirror images in the line y = x. WASSCE Tutor diagram.

Reading logarithm notation

• $\log_2 32 = 5$ is read "log to base $2$ of $32$ equals $5$."
• $\log_{10} 1000 = 3$ is read "log of $1000$ to base $10$ equals $3$" or just "log $1000$ equals $3$" (because base $10$ is the standard common log).
• $\log_5 125 = 3$ is read "log of $125$ to base $5$ equals $3$."
• $\ln a$ (with no subscript) is the natural log — base $e \approx 2.718$. WAEC focuses on common log (base $10$) and other integer bases.

When the base is not written, it is base $10$ in WAEC convention. So $\log 100 = 2$.

Converting between index and log forms

This is the foundational skill. Be able to convert in both directions.

Index → log

$2^4 = 16$ → $\log_2 16 = 4$. $5^3 = 125$ → $\log_5 125 = 3$. $10^{-2} = 0.01$ → $\log_{10} 0.01 = -2$. $4^{1/2} = 2$ → $\log_4 2 = \dfrac{1}{2}$.

Log → index

$\log_3 81 = 4$ → $3^4 = 81$. $\log_{10} 10000 = 4$ → $10^4 = 10000$. $\log_2 \dfrac{1}{4} = -2$ → $2^{-2} = \dfrac{1}{4}$. $\log_8 2 = \dfrac{1}{3}$ → $8^{1/3} = 2$.

Computing simple logarithms

For simple cases, ask yourself: "What power of $b$ gives $a$?"

Example 1

Find $\log_2 8$.

What power of $2$ gives $8$? $2^3 = 8$. So $\log_2 8 = 3$.

Example 2

Find $\log_5 125$.

$5^3 = 125$. So $\log_5 125 = 3$.

Example 3

Find $\log_{10} 10000$.

$10^4 = 10000$. So $\log_{10} 10000 = 4$.

Example 4

Find $\log_4 64$.

$4^3 = 64$. So $\log_4 64 = 3$.

Example 5

Find $\log_3 \dfrac{1}{27}$.

$3^{-3} = \dfrac{1}{27}$. So $\log_3 \dfrac{1}{27} = -3$.

Example 6

Find $\log_9 3$.

$9^{1/2} = 3$. So $\log_9 3 = \dfrac{1}{2}$.

Example 7

Find $\log_8 4$.

Think: $8^x = 4$, both are powers of $2$. $8 = 2^3$, $4 = 2^2$. So $2^{3x} = 2^2$, $3x = 2$, $x = \dfrac{2}{3}$. So $\log_8 4 = \dfrac{2}{3}$.

Example 8

Find $\log_2 1$.

$2^0 = 1$. So $\log_2 1 = 0$.

This is a key fact: $\log_b 1 = 0$ for any base $b$. (Because any base to the power $0$ is $1$.)

Example 9

Find $\log_5 5$.

$5^1 = 5$. So $\log_5 5 = 1$.

Another key fact: $\log_b b = 1$.

Key facts to memorise

These show up constantly. Know them by heart.

Fact 1: $\log_b 1 = 0$

Any base raised to the power $0$ is $1$. So $\log_b 1 = 0$ for any valid base $b$.

• $\log_2 1 = 0$
• $\log_{10} 1 = 0$
• $\log_{100} 1 = 0$
• $\log_{0.5} 1 = 0$

Fact 2: $\log_b b = 1$

Any base raised to the power $1$ is itself. So $\log_b b = 1$.

• $\log_2 2 = 1$
• $\log_{10} 10 = 1$
• $\log_5 5 = 1$

Fact 3: $\log_b b^n = n$

The log of a base to a power gives back the power.

• $\log_2 2^5 = 5$
• $\log_{10} 10^3 = 3$
• $\log_3 3^{-2} = -2$

Fact 4: $b^{\log_b a} = a$

A base raised to its own log gives back the argument.

• $2^{\log_2 16} = 16$
• $10^{\log_{10} 100} = 100$

These are the inverse properties: logs and exponentials undo each other.

Domain — what can go inside a log?

The argument of a logarithm must be positive.

• $\log_2 8$ is fine ($8 > 0$).
• $\log_2 0$ is undefined (you can never get $0$ by raising $2$ to a real power).
• $\log_2 (-8)$ is undefined (you can never get a negative number by raising a positive base to a real power).

Why? Because $b^x > 0$ for any positive $b$ and real $x$. So $\log_b a$ exists only for $a > 0$.

Also, the base must be positive and not equal to $1$.

• $b = 1$ is excluded because $1^x = 1$ for all $x$, so $\log_1 a$ would be impossible to define uniquely.
• $b \leq 0$ is excluded because then $b^x$ is not always defined for real $x$.

Why logarithms exist — historical context

Before calculators, multiplying two large numbers like $4598 \times 3147$ took ages. Logarithms made this easier because:

$$\log(a \times b) = \log a + \log b$$

So you'd:

1. Look up $\log 4598$ in a table (about $3.66$).
2. Look up $\log 3147$ in a table (about $3.50$).
3. Add: $3.66 + 3.50 = 7.16$.
4. Look up the antilog of $7.16$ — that's the product.

Much faster than direct multiplication. This is why the logarithm laws (next lesson) were transformative.

Today, calculators handle large multiplications instantly, but logarithms still matter for:

• Solving exponential equations (where the unknown is in the exponent)
• Modelling growth and decay (population, radioactive material, compound interest)
• Sound intensity (decibel scale uses logs)
• Earthquakes (Richter scale uses logs)
• pH in chemistry (uses log)
• Information theory (bits use log base $2$)

WAEC tests them because they're foundational, not because students still do hand calculations with log tables.

Common logarithm — base $10$

When the base is $10$, it's called the common logarithm. The notation often omits the base:

• $\log 100 = 2$ means $\log_{10} 100 = 2$.
• $\log 1000 = 3$ means $\log_{10} 1000 = 3$.
• $\log 0.1 = -1$.
• $\log 0.01 = -2$.

The next lesson shows how common logarithms are used in calculations.

WASSCE examination focus

Practice problems

1. Find $\log_3 27$.

2. Find $\log_{10} 1000$.

3. Find $\log_2 64$.

4. Find $\log_5 \dfrac{1}{25}$.

5. Find $\log_4 64$.

6. Find $\log_{10} 0.01$.

7. Find $\log_3 \sqrt{3}$.

8. Express $7^2 = 49$ in log form.

9. Express $\log_5 125 = 3$ in index form.

10. Find $\log_8 16$.

11. Find $\log_2 1$.

12. Find $\log_{10} 10^7$.

Worked solutions

1. $3^3 = 27$. Answer: $3$.

2. $10^3 = 1000$. Answer: $3$.

3. $2^6 = 64$. Answer: $6$.

4. $5^{-2} = \dfrac{1}{25}$. Answer: $-2$.

5. $4^3 = 64$. Answer: $3$.

6. $10^{-2} = 0.01$. Answer: $-2$.

7. $\sqrt{3} = 3^{1/2}$. So $\log_3 3^{1/2} = \dfrac{1}{2}$.

8. $\log_7 49 = 2$.

9. $5^3 = 125$.

10. $8 = 2^3$, $16 = 2^4$. $\log_8 16 = x$ means $8^x = 16$, i.e. $2^{3x} = 2^4$, so $x = \dfrac{4}{3}$.

11. $\log_2 1 = 0$ (any base to power $0$ is $1$).

12. $\log_{10} 10^7 = 7$ (using Fact 3).

What's next

In the next lesson we cover the laws of logarithms — the rules that turn multiplication into addition, division into subtraction, and powers into multiplication. These three laws are the heart of the topic and the source of most WAEC questions.