Logarithms
Extract from the book 'Digital Technology'
In 1614, John Napier, Baron of Merchiston, in Scotland, proposed a method of adding numbers to carry out multiplications. In his book, Mirifici Logarithmorum Canonis Descriptio, Napier tabled a series of numbers that, when added together, resulted in the multiplication of those numbers. Today these are called Logarithm Tables.
The use of logarithms is an important step in the computation of numbers and contributed to significant advances in science, navigation and astronomy.
How Logarithms work
Example using decimal numbers:
Take a number, say 1000 and divide it by 10, we get a result of 100. If we divide that number by 10 we get a result of 10. Conversely, if we multiply
10 x 10 x 10 we get 1000.
In logarithmic terms, 1000 = 103, that is, the logarithm to the base 10 of 1000 is 3. The 3 signify the number of times 10 needs to be multiplied by it to get 1000.
Another way of writing this is log10 (1000) = 3.
Other base values such as binary may be used. Binary uses two digits 0 and 1 for counting and computations.
An example using binary numbers:
If we multiply 2 x 2 x 2 x 2 x 2 we get a result of 32, that is, 2 multiplied by itself five times gives a result of 32.
In logarithmic terms 25 = 32.
Also 24 = 16.
Now 16 x 32 = 512.
If we add the binary logarithmic numbers together 25 + 24 we get a result of 29.
Now 2 multiplied by itself nine times equals 512.
So 29 = 512
Another way of writing this is log2 (512) = 9.
The result of adding the indices of the two logarithmic numbers together is equivalent to multiplying the binary numbers.
By placing appropriate markings corresponding to logarithmic numbers on a sliding ruler, multiplication and division computations can be speedily accomplished. The accuracy of the results, however, depends on the accuracy of the markings and the accuracy of the reading.
Extract from the book 'Digital Technology'