What is the value of $x$ when $a^\frac{1}{x}=1$?

Question

I used to compute complexity of an algorithm which reaches to constant value after x level because of $a^\frac{1}{x}=1$. Now I need to find $x$ to reach answer.

To describe more : my recursive algorithm is $T(n)=2T(\sqrt{n})+log_{2}^{n}$ ; ans $T(1)=1$. I tried to solve it using binary tree which in every level child node has sqrt of it's parent value.

Answer 1

Well, since $\frac 1x \log a = 0$, $a$ must be equal to $1$ and $x$ can be absolutely any real number whatsoever!

Answer 2

If $a =1$, any $x\neq 0$ will do. Otherwise, you would need $1/x=0$ which of course never happens (in the usual $\Bbb{R}$)

Answer 3

You could say $\lim_{x\to\infty}a^{1/x}=1$, but has been already noted there is no specific value for $x$ that results in $a^{1/x}$ equaling $1$.