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Useful Computational Methods

### Cube root via logarithms

Electronic calculators became publicly available around 1970 A.D. Before that, the method used for finding cube roots employed "common logarithms". In those days, bookstores used to sell booklets listing the logarithms and antilogarithms of numbers from 0001 to 9999. Common logarithms refer to logarithms where the base is 10. A rule of logarithms in any base b is that log b xt = t log b x, which holds for all real numbers t and positive real numbers b and x. Thus in common logarithms too, log 10 x1/3 = 1/3 log 10 x. So if x = 5, log 10 51/3 = 1/3 log 10 5. But the table of common logarithms gives log 10 5 = 0.6990 to four digits of precision so that log 10 51/3 = (1/3)(0.6990) = 0.2330. Now, to get the value of x, all we need to do is find the number whose common logarithm is 0.233. That is, we need the antilogarithm of 0.233. The table of commmon antilogarithms gives antilog (0.233) as ???

The example shows that we can find the cube root of any real number in this manner. However, since the table of logarithms or antilogarithms listed each number only in four digits, we were only able to get cube roots with precision of at most four digits using logarithms. Since the advent of calculators that also have a key for finding the antilogarithm, the precision came to be limited by one less than the number of digits given by the calculator.

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