Logarithmic Properties for Equation

Question

What are the steps/logarithmic properties used to solve the equation 2^n=n^8? Thanks!

Answer 1

Taking the natural log of both sides, $$\ln 2^n=\ln n^8 \\ n\ln 2=8\ln n \\ \frac{n}{\ln n}=\frac{8}{\ln 2} $$ This is a transcendental equation, meaning it cannot be solved exactly. There are two solutions, approximately equal to $1.1$ and $43.56$ according to Wolfram Alpha. You may also be interested in learning about the Lambert W function.

Answer 2

NotNotLogical gave you the way to approach the problem which leads to a transcendental equation in $n$. The two roots are given by $$x_1=-\frac{8 W\left(-\frac{\log (2)}{8}\right)}{\log (2)}$$ $$x_2=-\frac{8 W_{-1}\left(-\frac{\log (2)}{8}\right)}{\log (2)}$$

I a more general manner, equations such that $$a^n=n^b$$ have solutions given by $$x=-\frac{b W\left(-\frac{\log (a)}{b}\right)}{\log (a)}$$ which can potentially correspond to the two branches of Lambert function.

As Lucian nicely suggested, a look at function $\dfrac{\ln n}n$ is very useful since you should notice that there is no real solution if $\frac{\log (a)}{b} \gt e$ (this was established by Euler).