Variables and exponents

Question

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 1

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 $n\approx 204.047$.