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Name: Ralph C.
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Question:
How do you use logarithms to multiply or divide very large or very small numbers?



Replies:
Ralph,

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:
      log(1.457)=0.1635            log(7.938)=0.8997
      

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 non-integer numbers.

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
Physics Instructor
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".

Vince Calder



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