Question 78794
{{{ (e^3) * x^e * e^x^2 = e^10 }}}

This will turn out pretty easy or else impossible if you take the natural logarithm of both sides and then do something about the x^e and then solve with algebra.

The logarithm rules are:

1. {{{ log(a*b) = log(a) + log(b) }}}
2. {{{ log(e^c) = c }}}
3. {{{ log(a^b) = log(a) * b}}}

Use rule 1 first:
{{{ log(e^3) + log(x^3) + log(e^x^2) = log(e^10) }}}
Use rule 2 and 3:
{{{ 3 + 3 log(x) + x^2 = 10 }}}

{{{ x^2 - 7 + 3 log(x) = 0 }}}

I don't think there is a way to solve this algebraically.  A person can find numbers for x that work very closely, using graphical or numerical methods and tools.  I am sorry that I do not have a name for this type of equation.