Is $n^{1/\log \log n} = O(1)$? Is $n^{1/\log \log n} = O(1)$ ?


Suppose that $n^{1/\log \log n} = c$ where $c$ is constant.

Taking logs of both sides,

$$\frac{1}{\log \log n}\log n = \log c.$$

I am not able to spot an error. Please help
The function $n^{1/\log \log n}$ tends to infinity, since
$$
n^{1/\log\log n} = e^{\log n/\log\log n},
$$
and $\log n/\log \log n \longrightarrow \infty$.