Consider the equation that $$xy=\log{y}+1\text{.}$$ How does one prove that $y$ cannot be expressed explicitly in terms of $x$?
By the way, I do not know how the adverb "explicitly" is strictly defined, though likely it is generally understood. For example, I would call $y=\displaystyle\frac{x-1}{\sin^3{x+1}}$ an explicit expression.
To complement the comment by Thomas: you can express $y$ explicitly by $x$ via the Lambert W function. You have $$\begin{array}{lrl} & xy & = \log(y) + 1 \\ \Rightarrow\ & e^{xy}& =y\cdot e \\ \Rightarrow\ & -xe^{-1} &= -xye^{-xy} \\ \Rightarrow\ & W(-xe^{-1}) &= -xy \\ \Rightarrow\ & y & = -\tfrac 1x W(-xe^{-1}) \end{array}$$