304 ALGEBRA.
Thus the fraction
whence, by definition, log. =x—y= log, M—log, N.
Ex. log (2 1 7) = log 1 = log 15 — log7 = log (8 x 5)— log7 = log 3 + log 5 — log 7.
433. Logarithm of a Power. The logarithm of a number raised to any power, integral or fractional, is the logarithm of the number multiplied by the index of the power.
Let log, (Mp) be required, and suppose
x= loga M, so that ax= M; then Mp = (ax)p = apx whence, by definition, loga (Mp) = px; that is, loga (Mp) = p loga M.
Similarly, log a (M1 r) = 1 r loga M.
Ex. Express the logarithm of = in terms of log a, log b, and log c?
a ca 3 : log ME = log aos log a? — log (cb?) = $loga — (loge? + log b?) = 3 loga — 5 loge — 2 log b.
434. From the equation 10x = N, it is evident that common logarithms will not in general be integral, and that they will not always be positive.
For instance, 3154 > 10^3 and < 10^4;
.. log 3154 = 3 + a fraction.
Again, .06>10^-2 and <10^-1 -. log .06 =— 2 + a fraction.
Negative numbers have no common logarithms.
435. Definition. The integral part of a logarithm is called the characteristic, and the decimal part, when it is so written that it is positive, is called the mantissa.
The characteristic of the logarithm of any number to the base io can be written by inspection, as we shall now show.
