Question

Consider the equation $x^{\sqrt{x}}=(\sqrt{x}{)}^x,x>0$. Let $p$ and $q$ be the integral solutions. Then

• A. the sum of the solutions is $5$
• B. the solutions are coprime
• C. ${\log }_{p}q$ and ${\log }_{q}p$ are defined
• D. If ${\log }_{q}p$ is defined, then ${\log }_{p}q$ is not defined and vice-versa

Answer 1

Answer: (A), (B), (D)

If ${\log }_{q}p$ is defined, then ${\log }_{p}q$ is not defined and vice-versa

$x^{\sqrt{x}}=(\sqrt{x}{)}^x$
Clearly, $x>0$
Taking $\ln$ both sides
$\sqrt{x}\ln x=x\ln \sqrt{x}\Rightarrow \sqrt{x}\ln x-\frac{x}{2}\ln x=0\Rightarrow \ln x(2\sqrt{x}-x)=0$
$\Rightarrow \ln x=0$ or $2\sqrt{x}-x=0$
$\Rightarrow x=1$ or $x=4$

${\log }_{a}x$ is defined for $x>0,a>0$ and $a\ne 1.$
Hence, if ${\log }_{q}p$ is defined, then ${\log }_{p}q$ is not defined and vice-versa.