exponentiation - Variables and exponents

How would you solve this equation ?

$$500n=4000(1.016)^n$$

I tried using some logarithms but I could not do it. The only unknown variable is n but I'm having a bit of trouble getting there.

Answer

The practical way to solve such equations is to approximate the solution numerically. There's no expression for the solution in terms of elementary functions.

You can write down a solution explicitly using Lambert's W function as

$$n = - \frac{W(-\frac{4000}{500} \ln(1.016))}{\ln(1.016)}$$
but that's hardly enlightening since the $W(a)$ is itself defined as "the solution to $xe^x=a$", which has a similar form to your original equation.

Numerically, Wolfram Alpha finds the solutions $n\approx9.26787$ and $n\approx 204.047$.