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82 THE FASCINATION OF NUMBERS

between 10 and 100 and higher powers of 10 by converting
the intervening integers to approximate powers of 10. For
example, 100 equals 10% and the logarithm of 100 is there-
fore 2. In the same way, the number g is approximately
equal to 1047?! and the number 79 is approximately equal
to 1018976, 50 that the logarithms of 3 and 79 are -4771 and
1-8976 respectively. Emphasis is laid upon the fact that
although these equivalents are only approximate and not
exact, the margin of error is so small as to be negligible,
except where results to many more significant figures are
required.

There are two special points worthy of note. Firstly, the
logarithm of the base number itself is always 1, because
a'=a, and it follows that the logarithm of a is 1. Secondly,
the logarithm of 1 is always o, whatever base is used, for
a®=1, irrespective of the value of a.

At first sight the second example may seem absurd, since it
means that 1°=2°=39, etc., but examination of the meaning
of a° removes the apparent absurdity. For a° is the same as
a"~" and this is the same as 4" divided by a", and this is
clearly equal to 1 for any value of a. This will be more clearly
understood if we take a" to be 1 xa" (which is obviously the
same) and, instead of saying that a" represents the product
of a taken n times, we say that it represents the number 1
multiplied by a taken z times. This is a very different descrip-
tion. It follows then that @° being the same as 1 X a°, means
‘take the number 1 and do not multiply it by any number of
a’s at all’. In other words, the number 1 is not to be multi-
plied at all and it must therefore remain 1.

The use of logarithms for multiplication and division
reduces the working of the process considerably, especially
where there are a number of simultaneous operations. Their
use is, however, even greater in the extraction of roots, for
not only can this be done in a matter of seconds but, what is
more important, any root of any degree may be found with
equal ease. There is, for instance, no practical way of finding
the fifth root of a number by ordinary arithmetical pro-