Use of the logarithm

Addition and subtraction

In elementary school we learned how to add numbers: 2 + 3 = 5.
But we also learned how to answer the question: what must be added to 2 in order to get 5?
Or: 2 + ? = 5.
The answer is found by means of subtraction: ? = 5 - 2 = 3.
We could as well have asked: ? + 3 = 5.
Subtraction gives also the answer to that question: ? = 5 - 3 = 2.

Subtraction is called the inverse of addition.

Multiplication and division

In elementary school we learned how to multiply numbers: 2 × 3 = 6.
And we learned to answer the question: what must be multiplied by 2 in order to get 6?
Or: 2 × ? = 6.
In this case the answer is found by division: ? = 6 / 2 = 3.
We could as well have asked: ? × 3 = 6.
Again, the answer is obtained by division: ? = 6 / 3 = 2.

Division is called the inverse multiplication.

Power, root and logarithm

We probably learned about the power function in high school: 2³ = 2 × 2 × 2 = 8.
What must be raised to the power 3 in order to get 8? Or: ? ³ = 8.
The answer is the root function: ? = ³√ 8 = 2.
But the root function doesn't give the answer to the question: to what power must  2  be raised in order to get 8? Or: 2 ^? = 8.
To answer this question we need the logarithm: ? = ²log(8) = 3.

Therefore, there are two different inverses of the power function: root and logarithm.

Logarithm

The use of the logarithm is best illustrated by a simple example:     2³ × 2⁵ = 2⁸

Per definition we have:

    3 = ²log(2³)     5 = ²log(2⁵)     8 = ²log(2⁸) = ²log(2³ × 2⁵)

Note that 8 = 3 + 5, and therefore:

    ²log(2³ × 2⁵) = ²log(2³) + ²log(2⁵)

The general relation is ^clog(a × b) = ^clog(a) + ^clog(b). Thus the logarithm of the product equals the sum of the logarithms of the two factors.

The logarithm makes it possible to multiply with the aid of a ruler.

This property is applied when plotting data on paper with a logarithmic grid.