38 MATHEMATICAL THEORY OF INVESTMENT 16. The transformation of logarithms of one system into loga- rithms of another system. In theoretical work the so-called Napierian logarithms, whose base is the number e= 2.71828+, are almost invariably used, while for numerical computation the common logarithms, wliose base is 10, are used. It is therefore fre- quently necessary to transform a Napierian logarithm into a com- mon logarithm, and vice versa. This transformation is a special case of a more general problem, viz. to express a logarithm with a given base a in terms of a logarithm with another base b. Let JV" be the number whose logarithm is sought and I its logarithm to the base a. Then, by hypothesis, log^=?. (1) Changing equation (1) to the exponential form, N=a1. (2) Taking logarithms to the base b of both sides of equation (2), log^=nog,a. (3) If I be replaced by its value as given by equation (1), the result is log.-Z^ log.fli log<.Ar, (4) which is the fundamental formula in the problem of transform- ing the logarithms of one system into those of another system. If it be required to find the Napierian logarithm when the com- mon logarithm is given, we replace a by 10 and b by e= 2.71828 +. Equation (4) then becomes log^=log,10.1og^. (5) To find the common logarithm when the Napierian logarithm is given, solve (5) for log -ZV" and obtain ^^log^To-10^ (6) The factor ———, by means of which Napierian logarithms are log JO converted into common logarithms, is called the modulus of the common system and is usually denoted by the symbol M, so that by definition i "i^TO (7) LOGARITHMS 39 The number M and its reciprocal, ^I^IO, (8) have been carefully computed to a large number of decimal places. Their values to the sixth place are M=. 434294 and -1- == 2.302585. If the values of Jifand — be introduced into equations (5) and (6), these equations become log^=^log^= 2.302585 log^ (5') and log^JV=Jflog^=. 434294 log, IV. (6') From equations (5') and (6') the following rules for the trans- formation of logarithms of the one system into logarithms of the other system are easily obtained. To find the .Napierian logarithm when the common logarithm is given, multiply the common logarithm by 2.302585. To find the common logarithm when the Napierian logarithm is given, multiply the .Napierian logarithm by .434294.
EXAMPLES 1. The common logarithm of 2 is approximately .3010300. What is its Napierian logarithm ? 2. By means of a table of common logarithms and one of the formulas of the present section find the Napierian logarithm of 258. 3. Prove that log,,, e = ——— and that, in general, logs a = ——— Why ^ log,,10 log,. 6 do we write M = -——— rather than M = logjgg? 4. Compute the common logarithm of 1.1 to five places by starting from the power series 2 3 „-» log(l+^)=x-|-+|—^+....