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May 19th

May 19th

# The Story of a Number

COMPUTING WITH LOGARITHMS

For many of us - at least those who completed our college education after 1980 - logarithms are a theoretical subject,
taught in an introductory algebra course as part of the function concept. But until the late 1970s logarithms were still
widely used as a computational device, virtually un changed from Briggs 's common logarithms of 1624. The advent of
the hand-held calculator has made their use obsolete.

Let us say it is the year 1970 and we are asked to compute the expression

For this task we need a table of four-place common logarithms (which can still be found at the back of most algebra
textbooks). We also need to use the laws of logarithms :

where a and b denote any positive numbers and n any real number; here "log" stands for common logarithm - that is,
logarithm base 10 - although any other base for which tables are available could be used.

Before we start the computation, let us recall the definition of logarithm: If a positive number N is written as
then L is the logarithm (base 10) of N, written Thus the equations and are equivalent -
they give exactly the same information. Since and we have  and Therefore, the
logarithm of any number between 1 (inclusive) and 10 (exclusive) is a positive fraction, that is, a number of the form
in the same way, the logarithm of any number between 10 (inclusive) and 100 (exclusive) is of the form
and so on. We summarize this as:

(The table can be extended backward to include fractions, but we have not done so here in order to keep the
discussion simple .) Thus, if a logarithm is written as the integer P tells us in what range of
powers of 10 the number N lies; for example, if we are told that we can conclude that N lies
between 1,000 and 10,000. The actual value of N is determined by the fractional part . a b c . . . of the logarithm. The
integral part p of is called its characteristic, and the fractional part its mantissa. A table of
logarithms usually gives only the mantissa; it is up to the user to determine the characteristic. Note that two logarithms
with the same mantissa but different characteristics correspond to two numbers having the same digits but a different
position of the decimal point. For example, corresponds to , whereas
corresponds to . This becomes clear if we write these two statements in exponential form :
while

We are now ready to start our computation. We begin by writing x in a form more suitable for logarithmic computation
by replacing the radical with a fractional exponent:

Taking the logarithm of both sides, we have

We now find each logarithm, using the Proportional Parts section of the table to add the value given there to that given
in the main table. Thus, to find log 493.8 we locate the row that starts with 49, move across to the column headed by 3
(where we find 6928), and then look under the column 8 in the Proportional Parts to find the entry 7. We add this entry
to 6928 and get 6935. Since 493.8 is between 100 and 1,000, the characteristic is 2; we thus have
We do the same for the other numbers. It is convenient to do the computation in a table:

For the last step we used a table of antilogarithms-logarithms in reverse. We look up the number 0.5780 (the mantissa)
and find the entry 3784; since the characteristic of 1.5780 is 1, we know that the number must be between 10 and 100.
Thus x = 37.84, rounded to two places.

Sounds complicated? Yes, if you have been spoiled by the calculator. With some experience, the above calculation
can be completed in two or three minutes; on a calculator it should take no more than a few seconds (and you get the
answer correct to six places, 37.845331). But let us not forget that from 1614, the year logarithms were invented, to
around 1945, when the first electronic computers became operative , logarithms - or their mechanical equivalent, the
slide rule-were practically the only way to perform such calculations. No wonder the scientific community embraced
them with such enthusiasm. As the eminent mathematician Pierre Simon Laplace said, "By shortening the labors, the
invention of logarithms doubled the life of the astronomer."

NOTE

1. The terms characteristic and mantissa were suggested by Henry Briggs in 1624. The word mantissa is a late Latin
term of Etruscan origin, meaning a makeweight, a small weight added to a scale to bring the weight to a desired value.
See David Eugene Smith, History of Mathematics, 2 vols. (1923; rpt. New York: Dover, 1958),2:514.

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