LOGARITHMS. 355
436. The Characteristic of the Logarithm of Any Number Greater than Unity. It is clear that a number with two digits in its integral part lies between 10^1 and 10^2; a number with three digits in its integral part lies between 10^2 and 10^3; and so on. Hence a number with n digits in its integral part lies between 10^n-1 and 10^n.
Let N be a number whose integral part contains n digits; then
N= 10^{(n-1)+a fraction};
log N = (n-1) + a fraction.
Hence the characteristic is n-1; that is, the characteristic of the logarithm of a number greater than unity is less by one than the number of digits in its integral part, and is positive.
437. The Characteristic of the Logarithm of a Decimal Fraction. A decimal with one cipher immediately after the decimal point, such as .0324, being greater than .01 and less than .1, lies between 10^-2 and 10^-1; a number with two ciphers after the decimal point lies between 10^-3 and 10^-2; and so on. Hence a decimal fraction with n ciphers immediately after the decimal point lies between 10^{-(n+1)} and 10^n.
Let D be a decimal beginning with n ciphers; then
D= 10^{-(n+1)+a fraction};
log D = -(n+1) + a fraction. .
Hence the characteristic is -(n +1); that is, the characteristic of the logarithm of a decimal fraction is greater by unity than the number of ciphers immediately after the decimal point and is negative.
438. Advantages of Common Logarithms. Common logarithms, because of the two great advantages of the base 10, are in common use. These two advantages are as follows:
(1) From the results already proved it is evident that the characteristics can be written by inspection, so that only the mantissæ have to be registered in the Tables.
