Does the equation $x^e+e^x=e^n$ have any real solutions?

Let $x,n$ be positive integers. Consider the equation $$e^x+x^e=e^n$$ I would like to know whether this equation can have any positive integer solutions. Since $x^e \gt 0$, any solution must have $n \gt x$. Dividing by $e^x$, $$1+\left(\frac{x}{e}\right)^e=e^{n-x},$$ so $$n-x= \log\bigg(1+\frac{x}{e}\bigg)^e \\n-x = e \log\bigg(1+\frac{x}{e}\bigg) \\ n-x = \bigg(1+\frac{x}{e}\bigg)$$