Page:Elementary algebra (1896).djvu/371

LOGARITHMS. 853

PROPERTIES OF LOGARITHMS.

428. Logarithm of Unity. The logarithm of 1 is 0.

For a^0 =1 for all values of a; therefore log 1 = 0, whatever the base may be.

429. Logarithm of the Base. The logarithm of the base itself is 1. For a =a; therefore loga a=1.

430. Logarithm of Zero. The logarithm of 0, in any system whose base is greater than unity, is minus infinity.

For $a^{-\infty }={\frac {1}{a^{\infty }}}$

Also, since a^+\infty = \infty, the logarithm of +\infty is +\infty.

431. Logarithm of a Product. The logarithm of a product is the sum of the logarithms of its factors.

Let MN be the product; let a be the base of the system, and suppose

x= loga M, y = loga N; so that a^x = M, a^y = N

Thus the product MN= a^x \times a^y = a^x+y;

whence, by definition, log MN=x+y= loga M + loga N.

Similarly, loga MNP = loga M+loga N+loga P; and so on for any number of factors.

Ex. log 42 = log (2 \times 8 \times 7)= log 2 + log 3 + log 7.

432. Logarithm of a Quotient. The logarithm of a quotient is the logarithm of the dividend minus the logarithm of the divisor.

Let M N be the fraction, and suppose

$x=\log _{a}M,y=\log _{a}N$;

so that $a^{x}=M,a^{y}=N$.