1968 AHSME Problems/Problem 29

Problem

Given the three numbers $x,y=x^x,z=x^{x^x}$ with $.9<x<1.0$ . Arranged in order of increasing magnitude, they are:

$\text{(A) } x,z,y\quad \text{(B) } x,y,z\quad \text{(C) } y,x,z\quad \text{(D) } y,z,x\quad \text{(E) } z,x,y$

Solution

Seeing that we need to compare values with exponents, we think logarithms. Taking the logarithm base $x$ of each term, we obtain $1$ , $x$ , and $x^x$ . Because $0<x<1$ , $f(n)=\log_x(n)$ is monotonically decreasing, so the order of terms by magnitude in our new set of numbers will be reversed compared to the original set (i.e. if $a<b<c$ , then $\log_x(a)>\log_x(b)>\log_x(c))$ . However, the order of this set will be reversed again (back to the order of the original set) when we take the logarithm base $x$ a second time. After doing this operation, we find the values $0$ , $1$ , and $x$ , which correspond to $x$ , $y$ , and $z$ , respectively. Because $0.9<x<1$ , $0<x<1$ , and so, by the correspondence detailed above, $x<z<y$ , which yields us answer choice $\fbox{A}$ .

1968 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 28	Followed by Problem 30
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1968 AHSME Problems/Problem 29

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