Somebody asked this here:

Prove that an equation has no elementary solution

But so far there is no response. The little math I know I have learnt it myself so I dont have a big picture of things. I was wondering if there is a general rule for showing that a function has no analytical solution. For instance:

$$10=x+\ln(x)$$

How can I find the value of $x$? How can I show this is not possible with simple mathematics? Even with no simple math, there is some tool to do it?