TITLE: How do I solve for $x$ in $\ln(x)\ln(x) = 2 +\ln(x)$
QUESTION [6 upvotes]: How do I solve for $x$?
$$\ln(x)\ln(x) = 2 +\ln(x)$$

REPLY [5 votes]: $$\ln(x)\ln(x) = 2 + \ln(x) \iff (\ln (x))^2 - \ln (x) - 2 = 0\tag{1}$$
Let $t = \ln x$. Then $(1)$ becomes $$t^2 - t - 2 = (t - 2)(t + 1) = 0\tag{2}$$
So, $$t = \ln x = 2 \implies x = e^2\;\text{ or }\; t = \ln x = -1 \implies x = e^{-1} = \frac 1e$$