Does the limit $lim_{ntoinfty}left(x^n-1right)^{1/n}$ exist?

For $x$ given, what do you think about the following limit?

$$\lim_{n\to\infty}\left(x^n-1\right)^{1/n}.$$

What I tried and what are the problems that I am facing:

Let $f(x, n)=\left(x^n-1\right)^{1/n}$. We have:

$$\log f(x, n)=\dfrac{1}{n}\log\left(x^n-1\right)=\dfrac{1}{n}\log\left(1-x^{-n}\right)+\dfrac{1}{n}\log\left(x^n\right),$$

first, I do not know if I can apply the log or not? I guess $x$ must be real? and must be positive? what about complex?

Finally,
$$\lim_{n\to\infty}\left(x^n-1\right)^{1/n}=\log x.$$

Answer

Here is a down and dirty solution, for large $n$, (and assuming $x>1$, since otherwise how can you take $n$ th root),

$$\frac{1}{2}x^n \leq x^n-1 \leq x^n$$
so

$$\frac{1}{\sqrt[n]{2}}x \leq (x^n-1)^{\frac{1}{n}} \leq x$$
So
$$(x^n-1)^{\frac{1}{n}} \to x$$