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000 + 00 = 00000

Logarithms Explained


Logarithms - log tables - classic oldfashioned conjuring trick for doing hard sums like magic. Take two big numbers to be multiplied together, look up the magic code numbers in the log-table, add the numbers, look up the magic decode number from the anti-log table, and there's the answer!


(A quick note for those born in the 21st Century and after: A long time ago, there were no electronic calculators. To add up / multiply numbers manually required a lot of work, but to multiply/divide was much more difficult. Logs were a school technique where you could add up some magic numbers looked up in a table of figures, and mysteriously the decoded result would be the answer!)


Did you ever do logarithms? Did you ever wonder how the "magic" worked? Did anyone ever explain it? If you were born before a particular time ago, chances are it's: YES,YES and NO.


Here's how it works:

Logarithms can multiply/divide any numbers, usually things like 8472 x 6339, but suppose you were to multiply as an example 1000 x 100. Obviously this is easy to do in the head! But look at HOW you do it. You have THREE 0s and TWO 0s and you ADD them, get FIVE 0s, and so the answer is 100,000.

OK, so you add together how many 0s there are. But the cunning trick of logarithms is that someone found you could have HALF of a 0, or TWO AND A THIRD 0s, etc, and add them up like that and get sensible answers! Half a 0 is the SQUARE ROOT of 10 (because adding two of them makes ONE 0).

For any number you start with, there's a "HOW MANY 0s" number, which is usually some funny decimal. LOG(2) is 0.30103 (to five places). These are the numbers you look up in the log-tables. What this means is that if 2 was a number with 1--- and so-many 0s it would have 0.30103 of them. Sounds odd, but here's an example:

125 x 9972

125 . . . how many 0s has that got? If it was 100 it would be 2.00000 precisely

Log table reveals... LOG(125) = 2.09691

9972 ... how many 0s? Nearly FOUR. (if it was 10,000 it would have exactly four zeros)

Log table reveals... LOG(9972) = 3.99878

Add them together. 2.09691 + 3.99878 = 6.09569

The answer is going to be six-and-a-bit 0s. More than a million

Looking it up in the anti-log table... Anti-log(6.09569) = 1246493

So that's the answer to five decimal places. So, 1246500 +- a bit.

(Logarithms are accurate to so-many decimal places rather than absolutely precise. This is good for engineering calculations where figures are within a working tolerance)

Time taken to do the calculation: A lot less than by long-multiplication. (but not as fast as using a calculator!)


Was that explained well? Or was it as clear as mud? Do conjuring tricks of mathematics need explaining? It depends whether you prefer to be mystified or to be a magician.

Logarithms have other applications, for example deciBels are measured on a logarithmic scale. The Richter Scale (for measuring earthquakes) is also a logarithmic scale.

Yes, but what about Natural logarithms, log e ? And other log BASES? Can do! See log e

"Log Tables" were little books full of pages of tables of numbers. Typically a log-table book such as "Castle's Logs" would contain Sine tables as well, and all kinds of other data. It was hardly interesting bedtime-reading exactly, but then it was supposed to be reference rather than for reading through. I always thought the "physical constants" section was the most interesting, with such things as the mass of the earth, the electron charge, etc! Now a question often asked is: How did they first create these log tables of numbers, before computers? Well, shocking as it may seem, they did it by the very long-winded methods of manual calculation. There were a few clever shortcuts that made it easier, but it was still a vast amount of work. Historically, you can see it was done manually at first, as log tables can have a few small mistakes in them.

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Thanks and Well Done to Google for being sensible and starting to include this page again. Shame be upon Google for fowling up the page title! Note: Blekko gets this right! Even Bing gets this right! Yahoo lists the page but fowls up the title in a variant of the style Google does.