Limit using central limit theorem

Suppose that $\left(X_n\right)_{n\in\mathbb{N}}$ are independent identically distributed random variables each following uniform distribution on $\left[0,1\right]$. Apply Central Limit Theorem to determine the limit $$\lim_{n\to\infty}\mathbb{P}\left(\left(X_1\cdot X_2\cdot\cdots\cdot X_n\right)^{n^{-\frac{1}{2}}}e^{n^{\frac{1}{2}}}\in\left[a,b\right]\right),\quad a,b\in\mathbb{R}_+\,.$$

Thanks !

Taking logs and denoting by $S_n$ the sums $S_n := \sum_{i=1}^n \log X_i$ we must find the limit of

$$P\left (\frac{S_n}{n^{1/2}} + n^{1/2} \in [\log a, \log b] \right).$$

Now let $Y:= \log X$. We have

$$E(Y) = \int_0^1 \log x dx = -1$$

and

$$E(Y^2) = \int_0^1 (\log x)^2 dx = 2$$

whence

$$\mathrm{var} (Y) = 1$$

so that in the usual notation $\sigma(Y)=1$. Hence

$$P\left (\frac{S_n}{n^{1/2}} + n^{1/2} \in [\log a, \log b] \right)= P\left (\frac{S_n - (-n)}{1 \cdot n^{1/2}} \in [\log a, \log b] \right).$$

I have written this in the usual form required for the CLT. The variances being finite, and the variables in the sum iid we conclude that the limit is

$$\frac{1}{\sqrt{2\pi}}\int_{\log a}^{\log b} e^{-t^2/2}dt.$$