A course of pure mathematics, by G.H. Hardy.

350 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS

[IX

that satisfied by logx. For if y, = log x1, y2 log x2, so that
x1 = eyl, x2 = eY2, we have y/ + y/2= log xI + log x2 = log (x1x2) and
eyl+y2 = e log(xlx2) =,12 =.ey x ey2.
Examples LXXXVII. 1. If dx/dy=x then x=iKey, where K is a
constant.
2. There is no solution of the equation f(x +y)=f(x)() f) fndamentally
distinct from f(y) =ey. [For, differentiating the equation with respect to x
and y in turn, we obtain
f'(x+y)=f'(x)f(y), f'(x+y)=f(x)f'(/)
and so f' (x)/f(x) =f' (y)/f(y), and therefore each is constant. Thus if z=f(y),
dz/dy =z/A, or
y=A =A logz + B,
A and B being constants; so that z=e(Y-B)/A.]
3. Prove that (e"a- l)/x-a as x —O. [Applying the Mean Value
Theorem, we obtain eax- 1 =aeat where 0 < 4 < x.]
188. (3) The function ey tends to + co with y more rapidly
than any power of y, or
lim ya/ey = lim e-Yy = 0
as y -- + co, for all values of a, however great.
We saw that (log x)/xg -G 0 as x - + oo, for any positive value
of f3, however small. Hence, if a = 1/3, (log x)a/x 0 for any
value of a, however large. The result follows by putting x = ey.
From this result it follows that we can construct a 'scale of infinity'
similar to that constructed in ~ 184, but extended in the opposite directioni.e. a scale of functions which tend to + oo with x more and more rapidly.
This scale is
X, X2 X3I... ex, e2X... e,..., e,..., eex,... eee 5...,
where of course ex2,..., ee',... denote e(X2),..., e()....
The reader should try to apply the remarks made in ~ 184 and Exs. LXXXVI,
about the logarithmic scale, to this 'exponential scale' also. The two scales
may of course (if the order of one is reversed) be combined into one scale... loglogx,... logx,... x,... ex,... ee...
189. The general power ax. The function ax has been
defined only for rational values of x, except in the particular case
when a = e. When a is rational and positive, the positive value
of the power ax is given by the equations
ax = (e log a)x = ex log a
We take this as our definition of ax when x is irrational. Thus
10^ = eW2. log 10. It is to be observed that ax, when x is irrational,