# Logarithms

Calculating Machines --- Abacus --- Napier's bones --- Slide Rule --- Logarithms --- Calculator

In the 1970's calculators became cheap enough to be used by everyone. Before that, either we had to do long multiplication and long division on paper, or we used slide rules or logarithms. Log tables are not a machine, of course, but they are a device used for calculation.

To understand logarithms, you need to understand how the powers of a number work. On this page, we are looking at powers of ten.

Ten to the power of two or 102 = 100
Ten to the power of three or 103 = 100
Ten to the power of six or 106 = 1000000

The little number above and to the right of 10 is called the exponent. For example, 102 has exponent 2. If you add the exponents of ten, you multiply the powers of ten.

100 x 1000 = 102 x 103 = 102+3 = 105 = 100000

This means that you can multiply the numbers just by adding the exponents. It's not particularly useful for exponents when they are whole numbers. Most people can multiply 100x1000! However, this also works for exponents which aren't whole numbers. How can we have exponents which aren't whole numbers? Well, since we can add the exponents to multiply the numbers, we know that:

10.5 x 100.5 = 10(0.5 + 0.5) = 101 = 10

So 100.5 must be the square root of ten (3.1622). You can work out exponents for other numbers as well.

 2 = 10.301 3.1622 = 10.5 4 = 10.6021

Once you have done this, imagine all these exponents, or logarithms as we call them, laid out in a table. We want to multiply two numbers. All you need to do is look up the logarithms in the table, and add them to get the logarithm of the answer. This isn't much good by itself, of course, but you can now look through the log table until you can find this logarithm, to get its number.

Log tables can't give logarithms of every possible number - there are too many! But luckily we know that 2x4 is the same as 0.2x40, and a tenth of 20x4 and ten times 20x0.04, and so on. So the table only has to cover from 10 to 99.9 (or 1 to 9.99, or .1 to .999). Then we have to work out how many extra zeroes we need, or where the decimal point goes. The table below covers from 1 to 9.99, as all the logarithms are less than one. If you want to multiply two numbers in their hundreds, then each logarithm needs 2 added to it (meaning multiply by 102 or 100). When you look up the logarithm, you only look up the part after the decimal point. The part before the decimal point tells you how many times you need to multiply (or divide) the answer by ten.

Here is a simplified log table for logs base 10, with examples below. The three pages are side by side to see them all at once.

We want to multiply or divide two numbers. Click on buttons to get their logarithms, then to add them together, then to look up the resulting logarithm to get the (approximate) answer. You can get more numbers if you wish.

You can see that you only get approximate answers using log tables. These simplified tables give three significant figures.

You can also use log tables to divide. You subtract the logarithms rather than adding them.

There are also anti-log tables, or inverse log. This means that you can look up the number from the logarithm directly, rather than having to search for it in the log table.

This is what real log tables looked like. The columns on the right of the page were used to get an extra figure of precision.