John Napier discovered the concept of logarithms.
A rule of logarithms in any base *b* is that log _{b} *N*^{t} = 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} *N*^{1/2} = 1/2 log _{10} *N*. So if N = 5, log _{10} 5^{1/2} = 1/2 log _{10} 5.
But, from calculator, log _{10} 5 = 0.6990... to some digits so that log _{10} 5^{1/2} = (1/2)(0.6990...) = 0.3495...

Now, to get the value of 5^{1/2}, 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: Antilog_{10} x = 10^{x}. So antilog(0.3495...) = 10^{0.3495...}, 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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