LOGARITHMS. By shortening processes of computation, logarithms have doubled the working speed of astronomers and engineers. The explanation of this highly practical branch of mathematics begins most conveniently with the exponential ex pression bi which represents some number n, so that we may write n = bl. In the elementary theory we assume b, n, 1 to be real numbers, b and n positive, and b greater than unity. If we know b and the exponent 1, we may compute n. For example, if b=io, 1=3, then n= r,000. If any two of the three numbers b, 1, n are given, the third may be computed. To develop the elementary theory of logarithms we assume the number n and the base b to be given, then the exponent 1 may be found. For the case under consideration (when b and n are both positive and b is greater than unity), the actual existence of one and only one real value I can be established by the modern theories of irrational numbers, such as that of Dedekind. For, if we make a "cut" of all ordered real numbers by placing in one class every number a for which ban, and in the other class every number (3 for which
> n, then if N is the number defined by the "cut," we must have
for the reason that every other assumption leads to an evident contradiction. That is, if we assume that
b is positive and greater than r.
Of all possible values greater than r which might be chosen as the base, two have been selected, which yield two systems of logarithms. One value is b= ro, used in the common logarithms, chosen because of certain practical advantages that accrue from a base which is the same as the scale of our number system. Common logarithms have also been called Briggian logarithms.
The other value taken for the base is 2.718 : it is usually represented by the letter e, and belongs to a type of irrational numbers called "transcendental numbers." The logarithms hav ing the base e are called natural logarithms, or sometimes less appropriately hyperbolic or Naperian logarithms. The base io yields logarithms that are most convenient for the purposes of computation; the base 2.718 yields logarithms which lead to simpler formulas in higher analysis than other systems, and are therefore the most "natural" ones to use.
According to the definition of logarithms given above, log arithms are really exponents and therefore are endowed with the properties of exponents, viz., b/ bu, = (i) where 1 and may be positive or negative, rational or irrational real numbers. In the theory of logarithms, if bi= n, we use the notation 1= logbn, and we read this, "1 is the logarithm of n to the base b." As b°= 1 and bi=b, it is evident that no matter what value different from zero the base may have, the logarithm of is zero, and the logarithm of the base itself is unity. Writing n and the three exponential formulas (I) translated into the notation of logarithms, become logs nni, / logs 111= logs (2) Substituting logbn for 1, for we have logbn = (3) Stated in words : (I) The logarithm of a product is equal to the sum of the logarithms of the factors. (2) The logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. (3) The logarithm of a power of a number is equal to the product of the exponent and the logarithm of the number itself.