Log problem

Algebra Level 4

$\Large\log_5x=x^{\left[5^{-1}(1+x^{-5^{-1}})\right]}5^{-5^{-1}}\qquad,\qquad x = \ ?$

Solve for positive integer $x$ .

1 solution

Ben Habeahan
Sep 3, 2015

Let $x=p^5$ or $p=x^{ \frac{1}{5}}.$

After substitute we have, $\log_{5}{p^5}={p^5}^{ [\ \frac{1}{5}(1+p^{-1})]\ }5^ \frac{-1}{5}.$

$\Leftrightarrow {((5)(5^ \frac{1}{5}))} \log_{5}{p}={p}^{(1+ p^{-1})}$

$\Leftrightarrow \log_{5}{p^{5^{ (\frac{6}{5})}}}={p}^{(1+\frac{1}{p})}$

$\Leftrightarrow p^{6}= 5^{p^{(1+\frac{1}{p})}}$

$\Leftrightarrow p^{6}= 5^{(p+1)}$

With $\color{#3D99F6}{observation},$ the statement $p^{5+1}= 5^{(p+1)}$ true iff $p=5.$

$x=p^5=5^5= \boxed{3125}$