 Solution

At a single point of contact the line is a tangent line, so its slope equals the derivative of the exponential function. At the contact coordinate $x_0$,Since $m>0$, dividing gives $x_0=1$, hence $m=e$. To verify uniqueness, the difference $e^x-ex$ has positive second derivative everywhere and its unique minimum is zero at $x=1$. Thus this line touches the curve once.