Chapter 1, Problem 50RE

(a) Show that for $x>0$ and $k\ne 0$ the equations $x^k=e^x$ and $\frac{\ln x}{x}=\frac{1}{k}$ have the same solutions.

(b) Use the graph of $y=\left(\ln x\right)/x$ to determine the values of $k$ for which the equation $x^k=e^x$ has two distinct positive solutions.

(c) Estimate the positive solution (s) of $x^8=e^x$ .