Page:Elementary algebra (1896).djvu/375

LOGARITHMS. 357 Ex. 1. Required the logarithm of .0002432.

In Seven-Place Tables we find that 3859636 is the mantissa of log 2432 (the decimal point as well as the characteristic being omitted); and, by Art. 437, the characteristic of the logarithm of the given number is -4;

log .0002432 = 4.3859636.

This may be written 6.5859636 - 10.

Ex. 2. Find the value of 5 .00000165, given log 165 = 2.2174889, log 697424 = 5.8434968.

Let x denote the value required; then

log x = log (.00000165)^{1 5} = {1 5} log (.00000165) = 1 (6.2174839);

the mantissa of log.00000165 being the same as that of log 165, and the characteristic being prefixed by the rule.

Now {1 5}(6.2174839)=1(10 + 4.2174839) = 2.8454968

and .8434968 is the mantissa of log 697424; hence x is a number consisting of these same digits, but with one cipher after the decimal point. [Art. 457.]

Thus x = .0697424,

441. Logarithms transformed from Base a to Base b. Suppose that the logarithms of all numbers to base a are known and tabulated.

Let N be any number whose logarithm to base b is required.

Let y = logb N, so that b^y = N;

= loga (b^y)= loga N;

that is, y loga b = loga N;

y = 1 loga b \times log a N

or logb N= {1 loga b} x loga N;

Now since N and b are given, loga N and loga b are known from the Tables, and thus logb N may be found. Hence to transform logarithms from base a to base b we multiply them all by {1 loga b} ; this is a constant quantity, and is given by the Tables; it is known as the modulus.