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?
Yes there are no algebraic solutions to this problem frankly I can't answer the last bit of your question however I can demonstrate an algebraic-ish solution. So basically there is something called Lambert W function and the function is shown by a capital W.
$W(c)=x , c= xe^x$
It is a pretty useful function which allows us to solve equations like $x^x=5$. Even though it is not an algebraic solution it still gives us an insight about what we're dealing with.
In the given example we can apply the following like this: $10-lnx=x$
$lne^{10}-lnx=x$
$ln{e^{10} \over x}=lne^x$
${e^{10} \over x} = e^x$
$e^{10} =xe^x$
$W(e^{10})=x\approx 7.9294$
You can find the values for the given input on Lambert W Function by using Wolfram Alpha http://www.wolframalpha.com/input/?x=0&y=0&i=productlog(e%5E10)