Logarithms

is a constant factor sometimes called the

modulus of the common system of logarithms. For the special case when n = b, we obtain I = logbib • Suppose that we know the common logarithm of 200 and we wish to compute the natural logarithm of 200. We have from the above 4- = 2.30103 • • • 4-0.43429 • • • = 5.2983 • • • , the natural logarithm of 200.

Logarithms of Complex Numbers.—If, in the equation bl = n, b is a positive real number and 1 is either positive or nega tive, but real, then n must necessarily be positive, for a positive number raised to a power that is real, though either positive or negative, always yields a positive result as its principal value. Under these restrictions a negative number has no logarithm. This limitation causes no embarrassment in computation with logarithms, for we proceed as if all factors were positive. If the number of negative factors is odd, we mark the final result as negative. From the standpoint of theory, however, the failure of the elementary exposition to include the logarithm of nega tive numbers indicates a lack of generality. It is found that, as soon as we drop the limitation that 1 shall be real, and permit / to become imaginary or complex, any number n, whether posi tive or negative, or even complex, has not only one logarithm, but an infinite set of them. To establish this fact we make use of two well-known theorems in trigonometry. One theorem states that the periodicity of the sine and cosine functions is emir, where m is any integer. The other theorem, due to Cotes and

Euler, is expressed by the formula - - where i=-4— i and 0 is any angle measured in radians. Accord ingly we obtain, That is, e has one real logarithm, namely 1, all the others are imaginary. Nate that the integer m may take also negative values. The conclusion reached is that in this more general treatment of logarithmic theory, every number has an infinite set of log arithms which are all imaginary except when the number n is positive, in which case there is one real logarithm to the base e which is the same as the natural logarithm obtained by the more restricted theory.

"Gaussian" Logarithms.

When a and b are known, there is often great need of a process for finding logio(a+b) and b), without making it necessary to pass from log a and log b to a and b. To achieve this expeditiously the Italian physi cist Zecchini Leonelli in 1803 suggested a new type of tables. A five-place table of this type was published in 1812 by Carl Friedrich Gauss hence the name "Gaussian logarithms." Six place tables of this kind are due to Karl Bremiker, to Siegmund Gundelfinger and to George W. Jones. A seven-place table was brought out by T. Wittstein. Leonelli and Gauss used a table consisting of three columns of figures, which Gauss marked by the letters A, B, C. Gauss lets A= logx, B= log(' C =log(' where x is a number greater than 1. It is seen that