First slide

Types of real valued functions XI

Question

If $p, q \in N$ satisfy the equation $x^{\sqrt{x}} = (\sqrt{x})^{x}$ then

Moderate

A

p+q-=5
B

$| p - q | = 4$
C

pq=4
D

if $\log_{q} p$ is defined, then $\log_{p} q$ is not and vice versa

Solution

$\begin{array}{l} x^{\sqrt{x}} = (\sqrt{x})^{x}, p, q \in N \\ \Rightarrow \sqrt{x} \log x = xlog \sqrt{x} \\ or \log x [\sqrt{x} - \frac{x}{2}] = 0 \\ \Rightarrow \log x = 0 or [\sqrt{x} - \frac{x}{2}] = 0 \\ \Rightarrow x = 1 or 4 \end{array}$

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