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**Logarithms**

Slide rules are based on the concept of logarithms. Although a knowledge of logarithms it is not necessary to be able to use a slide rule, it does help you to understand what you are doing.

If you multiply a number raised to two different powers, the result is equal to the same number raised to the sum of the powers.

A very simple case would be:

2^{2} x 2^{3}
= 2^{(2+3)} = 2^{5}

That is:

4 x 8 =
32

Another, slightly more general, example might be:

10^{a} x 10^{b} =
10^{(a+b)}

This suggests a way in which multiplication, difficult with complicated numbers, can be
replaced by addition, much easier even with complicated numbers. If "a" is
chosen so that 10^{a} equals the first number to be multiplied and
"b" so that 10^{b} equals the second number to be multiplied then all
that is needed is a way to work back and find out the number equal to 10^{(a+b)}.
If, say, one wanted to multiply 2 and 3 then "a" would have the value of
0.301 (since 10^{0.301} equals 2) and "b" would have the value 0.477
(since 10^{0.477} equals 3). The value of 0.301 in known as the logarithm of 2 and
the value of 0.477 as the logarithm of 3. The result of the multiplication is 10^{0.778}
(that is 10^{(0.301+0.477)} ) which in turn is equal to 6. The value 6 is known as
the antilogarithm of 0.788. Incidentally, the above example uses 10 raised to different
powers; other numbers could be used as the base but 10 was the most common.

A similar procedure can be used for division, except that the powers are subtracted.

Before calculators, tables of logarithms and antilogarithms were published. As can be imagined their use was cumbersome and prone to error. Examples of log tables are given here.

A slide rule gets round this complication by making the distances on the scales proportional to the logarithms of the numbers marked on the scales. By positioning two scales, one on the slide and one on the stock, in such a way that you are adding the distance of one scale to another, you are effectively adding the logarithms of these numbers and therefore multiplying them. Similarly, by positioning the scales to get the difference in distance between two numbers, you are effectively dividing them.