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Useful Computational Methods

Square-roots via logarithms

John Napier discovered the concept of logarithms. A rule of logarithms in any base b is that log b Nt = t log b N, which holds for all real numbers t and positive real numbers b and N. Thus in common logarithms too, log 10 N1/2 = 1/2 log 10 N. So if N = 5, log 10 51/2 = 1/2 log 10 5. But, from calculator, log 10 5 = 0.6990... to some digits so that log 10 51/2 = (1/2)(0.6990...) = 0.3495...
Now, to get the value of 51/2, all we need to do is find the number whose common logarithm is 0.3495... That is, we need the antilogarithm of 0.3495...
Again, the calculator would give commmon antilogarithms; antilog (0.0.395...) = ???
Another way is this: Antilog10 x = 10x. So antilog(0.3495...) = 100.3495..., which, again, you can find on the calculator.

The example shows that we can find the square-root of any real number in this manner. However, since the calculator gives logarithms or antilogarithms only in a few digits, we are only able to get square-roots with a precision determined by the number of digits seen on the calculator screen.

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