Maybe you mean

$$\\lnx=log_x(e^4)\\\\
x^{lnx}=x^{log_x(e^4)}\\\\
x^{lnx}=e^4\\\\
ln(x^{lnx})=ln{e^4}\\\\
(lnx)(lnx)=4lne\\\\
(lnx)^2=4\\\\
lnx=2\;\;or\;\;lnx=-2\\\\
x=e^2\;\;\;or\;\;\;x=\frac{1}{e^2}$$

$$\\log x = logx \times (e^4)\\$$

there is no soln to this

-------------------------------

Perhaps you mean

$$\\logx = log[x (e^4)]\\
logx = logx +log(e^4)\\$$

No - still no solutions