Page:Elementary algebra (1896).djvu/370

 CHAPTER XXXVIII.

LOGARITHMS.

425. Definition. The logarithm of any number to a given base is the index of the power to which the base must be raised in order to equal the given number. Thus if a^x = N, x is called the logarithm of N to the base a.

Examples. (1) Since 3^4 = 81, the logarithm of 81 to base 8 is 4.

(2) Since 10^1 = 10, 10^2=100, 10^3 = 1000,

the natural numbers 1, 2, 3, ... are respectively the logarithms of 10, 100, 1000, ... to base 10.

426. The logarithm of N to base a is usually written loga N, so that the same meaning is expressed by the two equations

a^x = N; x=log N.

Ex. Find the logarithm of 32 5 4 to base 2 2.

Let x be the required logarithm; then, by definition,

(2/2)* = 825/4; (2-22) = 29.295;

hence, by equating the indices, 34 = %; Oo ep OGG),

427. When it is understood that a particular system of logarithms is in use, the suffix denoting the base is omitted. Thus in arithmetical calculations in which 10 is the base, we usually write log 2, log3, ... instead of log10 2, log10 3, ...

Logarithms to the base 10 are known as Common Logarithms; this system was first introduced in 1615 by Briggs, a contemporary of Napier the inventor of Logarithms. 352