How to prove that $x^{\frac{1}{\log_a(x)}}$ is a constant? Are the following lines a valid proof that $x^{\frac{1}{\log_a(x)}}$ is a constant?
$$K = x^{\frac{1}{\log_a(x)}}$$
$$\log_a(K) = \frac{\log_a(x)}{\log_a(x)} = 1, \text{ for } x \notin \{0, 1\}$$
$$K = a$$
If not, how does one prove this?

The proof looks valid to me. I used natural logarithms instead of logarithms with base $a$, which makes the proof slightly longer:
$$ K=x^{\frac{1}{\log_ax}} \\ \\
\begin{align}
\ln K &=\ln\left(x^{\frac{1}{\log_ax}}\right) \\ 
&=\frac{1}{\log_ax}\ln x \\ \\
&=\frac{1}{\frac{\ln x}{\ln a}}\ln x \\ \\
&=\frac{\ln a}{\ln x}\ln x \\ \\
&= \ln a
\end{align} $$
and, as such, $K=a$. Your proof is much better, though!
