$\{U_j\}_{j\ge1}$ a sequence of independent and identically distributed. Prove convergence almost surely

Question

Let $\{U_j\}_{j\ge1}$ a sequence of independent and identically distributed random variables such that $U_1$ is a uniform random variable on $[0,1]$. Show that
$$X_n:=\Big(\prod_{j=1}^nU_j\Big)^{1/n}\rightarrow1/e$$
almost surely.

Since $U_1$ is uniform in [0,1] then the expected value is finite. My idea is to define a new sequence of random variables and apply the strong law of large numbers

Answer 1

$\log(X_n)=\frac{\sum_{j=1}^n \log(U_j)}{n}$. However $E\left(\log(U_j)\right)=\int_0^1\log(x)dx=-1$, so sum converges to $-1$ and $X_n\to e^{-1}$.