Computation And Mensuration

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(J) Any number in the decimal notation may be written as a common fraction whose numerator is integral and whose denominator is some power of 10. For example, 375.485 = ^Yrm ^ ^''^^^^ ^ ^^'^• Applying the rule just proved to this product, log 375.485 = log 375485 + log lO-s.
As a consequence of (J) the direct computation of the logarithms of whole numbers only is necessary.
As a consequence of (a) the direct computation of the logarithms of prime numbers only is necessary.
The logarithms to bas
...e 10 of the prime numbers may be computed to any required degree of approximation by using the infinite series (1) logio(l + x-) = logio X + 2(0.43429448) ' +..A^H-.-.:.JU^ + -..+ 2a; + l + 3(2a;+ 1)3 5(23^ + 1)5 7(2a;+l)7 1 + 1 (2 w - 1) (2 a; + 1)2 »-i (2n + l)(2x+ l)2«+i which is derived in the Differential Calculus.
...], COMPUTATION AND USE OF LOGARITHMS 47 The application of this formula requires the log^g x, that is, the logarithm of the number one less than the given prime, to be known.


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