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

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Perhaps you mean

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

No - still no solutions