Logs and Large Number Operation
Name: Ralph C.
How do you use logarithms to multiply or divide very
large or very small numbers?
The advantage of logarithms is two-fold. The major advantage is
log(A*B)=log(A)+log(B). The second advantage is that logarithms convert
easily to and from scientific notation (A*10^N), so long as "A" is between 1
and 10. It is important to use common logs(base 10) rather than natural
logs(base e) because natural logs do not convert easily. By the way, an
often seen way to express scientific notation in print and on calculators is
something like 1.45E7. This is much easier to type than 1.45*10^7 but means
the same thing.
Suppose we have two large numbers to multiply, perhaps 145723*7938475.
Convert to scientific notation: (1.45723E5)*(7.938475E6). The exponent is
the whole number part of the log, the part before the decimal point. The
part that is not the exponent (the front piece between 1 and 10)converts to
the fractional part of the log, the part after the decimal. Look it up on
the appropriate log table, or on a slide rule:
Our logarithms are now 5.1635 and 6.8997. You can add them on paper. The
sum is 12.0532. To convert back to standard numbers, the exponent will be
12. Us a table to covert 0.0532 back. It is the same as calculating
10^(0.0532). The value is 1.1303. Our product is now 1.1303E12. This same
method works for very small number, such as 0.00000456. The only difference
is the exponent, and as a result the logarithm, are negative.
This is how it is done without a calculator. This is how it was done for
many years. Before slide rules, people used tables in text books or on
cards. Slide rules used the principle of adding lengths. A length of
log(A) added to a length of log(B) produced a length of log(A*B). The
scales printed on the slide rules were already logarithmic. The conversions
to and from logarithms were effectively done automatically. I don't know
for sure, but I expect some calculators and computers use this technique for
With a calculator, you can do the conversions with the LOG button and the
10^x button. If you can find a log table, I think the "by hand" method
gives a much clearer picture of one of the original benefits of logarithms.
It is math history as well as mathematics.
Dr. Ken Mellendorf
Illinois Central College
A logarithm is an exponent of some number "B" called the base. That is,
"B" raised to the power "L" equals the number, "N". Written out as an
equation, this statement reads:
B^L = N, where the symbol "^" means the power or exponent to which the
number "B" is raised. Commonly the base number "B" is a positive, real
number. Even more commonly the number "B" = 10. So, 10^0 = 1; 10^1 = 10;
10^2 = 100; 10^3 = 1000; 10^4 = 10,000 and so on. Here the powers, "L" are:
0,1,2,3 and 4. The numbers, "N" are: 1, 10, 100, 1000, and 10,000. The
number "L" is called the logarithm of "N" to the base 10.
The power of "B" need not be an integer. For example: 10^1.5 = 10^3/2 or
verbally, " The square root of ten
cubed.". The power and utility of logarithms is that multiplication can be
carried out by the ADDITION of the exponents (which is usually easier than
multiplying). Example: 10^3 x 10^1 = 10 ^(3+1) = 10^4. Similarly, division
can be carried out by the SUBTRACTION of the exponents. Example: 10^4 / 10^3
= 10^(4-3) = 10^1.
To summarize then, the logarithm of a number "N" to the base "10" written:
log(N) is the power that the base, commonly the base is 10, is raised to
that equals "N".
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Update: June 2012