Problem: Given the three numbers x,y=xx,z=x(xx), with .9<x<1.0. Arranged in order of increasing magnitude, they are:
Answer Choices:
A. x,z,y
B. x,y,z
C. y,x,z
D. y,z,x
E. z,x,y
Solution:
Method I. Since 0<x<1, x to any positive power is less than 1 .
∴yx=x1−x<1 so that x<y,yz=xy−x<1 so that z<y, and zx=x1−y<1 so that x<z.∴x<z<y.
Method II. Since 0<x<1,logx<0. It follows that logx<xlogx or logx<logxx so that x<xx=y. Since z=x(xx)=xy,logz=ylogx, and, since y>x, ylogx<xlogx or logz<logy, so that z<y. However, since 0<y<1 and logx<0,ylogx>logx or logz>logx, so that x<z. Finally x<z<y.