Logarithms were not originally invented simply as another mathematical operation. They were developed as a computational shortcut for simplifying difficult arithmetic, especially multiplication, division, powers and roots.
1. Three Related Mathematical Questions
Consider the equation:
2³ = 8
The same relationship between 2, 3 and 8 leads to three different questions.
Power
A power asks for the value.
Question:
What is the value of 2³?
Answer:
2³ = 2 × 2 × 2 = 8
Therefore, the value is 8.
Root
A root asks for the base.
Question:
What is the base if the value is 8 and the exponent is 3?
Answer:
∛8 = 2
Therefore, the base is 2.
Logarithm
A logarithm asks for the exponent.
Question:
What is the exponent in 2ˣ = 8?
Answer:
log₂ 8 = 3
Therefore, the exponent is 3.
The Key Relationship
POWER → asks for the VALUE
ROOT → asks for the BASE
LOGARITHM → asks for the EXPONENT
This gives us three different ways of looking at the same relationship:
2³ = 8
∛8 = 2
log₂ 8 = 3
The logarithm therefore gives us the exponent needed to produce a given number from a specified base.
2. The Problem Mathematicians Faced
Before calculators and computers, astronomers, navigators and scientists had to perform long calculations by hand.
Multiplying large numbers repeatedly was tedious and prone to error.
For example, calculating a product such as
2345 × 6789
by ordinary multiplication requires several steps.
Mathematicians looked for ways of transforming difficult calculations into simpler ones.
They noticed an important relationship between multiplication and addition of exponents.
For powers with the same base:
aᵐ × aⁿ = aᵐ⁺ⁿ
For example:
2³ × 2⁴ = 2³⁺⁴
= 2⁷
This means multiplication of powers becomes addition of their exponents.
This relationship became the foundation for using logarithms as a computational tool.
3. Why Logarithms Were Useful
Suppose two numbers are expressed as powers of the same base:
A = 10ᵐ
B = 10ⁿ
When we multiply them:
AB = 10ᵐ × 10ⁿ
Therefore:
AB = 10ᵐ⁺ⁿ
Instead of multiplying the original numbers directly, we add their exponents.
Logarithms provide a systematic way of finding these exponents.
For example:
10² = 100
Therefore:
log₁₀ 100 = 2
Similarly:
10³ = 1000
Therefore:
log₁₀ 1000 = 3
The logarithm tells us the exponent.
4. The Definition of a Logarithm
If
bˣ = y
then
log_b y = x
where:
b is the base
x is the exponent
y is the result or argument of the logarithm.
Thus:
POWER FORM:
bˣ = y
LOGARITHMIC FORM:
log_b y = x
The two forms express the same mathematical relationship.
5. Examples
Example 1
Write 2⁵ = 32 in logarithmic form.
Since the exponent is 5:
log₂ 32 = 5
Example 2
Write log₃ 81 = 4 in exponential form.
Therefore:
3⁴ = 81
Example 3
Find log₁₀ 1000.
We ask:
10 to what power gives 1000?
10³ = 1000
Therefore:
log₁₀ 1000 = 3
Key Idea
A logarithm answers the question:
“What exponent is required?”
For example:
log₂ 16 = 4
because:
2⁴ = 16
Therefore, whenever you see a logarithm, think:
“What power of the base gives the number?”
By John Mutethia