CHAPTER XV. 



LOGARITHMS. 



I. After the extension of the theory of indices in Chapter XI so 

 as to embrace incommensurable exponents, we are enabled to give 

 an interpretation to the expression 



a" 

 for all possible values of x, integral or fractional, commensurable 

 or incommensurable. Since x appears in this expression in such 

 an unrestricted form it is common to speak of the expression as an 

 exponential function of x, intending to call attention thereby to the 

 fact that X may be considered a continuous variable as in an\' 

 ordinary algebraic function. 



If in the equation 



^^ = r, 

 we assume x to pass from one extreme of the algebraic scale to 

 the other, taking in every possible value, then we are able to give 

 a meaning to this equation in two variable ; because for every pos- 

 sible value of X, «', that is, r, has a definite meaning and value. 



In this connection it must be remembered that we are using a 

 and a^ inider the restrictions mentioned in XI, Art. 15. So that 

 when it'e speak of a"" we mean that a is a positive number, a?id by the 

 value of a" lue fnean that one of its values which is positive. Hence 

 in the equation «' = r we are to think of but one value of y re- 

 sulting w^hen any particular value is assigned to x. Thus in 

 io*^5=y^ve are to understand r= H-'v^ 10 and not r= — >^io or any 

 other possible value of y. 



Of course the very restrictions just mentioned prevent y from 

 having a negative value. Moreover, it is net evident that y can 

 have every positive value we please. For example, is it not plain 

 that a value of j>; exists which satisfies the equation 10* = -. In 

 general, while it is easy to see that in the equation 



there always exists a value of y for any value assigned to x, it is 

 far from evident that there exists a value of x corresponding to 

 every value which may be assigned to r. Whence the necessity 

 for the following theorems. 



