Solving inverse of function

Question

Given:

$f(t)=500(1.1)^t$

How would I mathmatically solve for:

$f^{-1}(3000)$

Answer 1

An easier suggestion: \begin{align*} &3000=500(1.1)^{t} \Longrightarrow\\ &6 = (1.1)^{t} \Longrightarrow\\ &\ln{6} = t \ln{1.1} \Longrightarrow\\ &t = \frac{\ln{6}}{\ln{1.1}}\Longrightarrow\\ \end{align*} So if we have the coordinate pair $\left(\frac{\ln{6}}{\ln{1.1}}, 3000 \right) \in f$, then
$\left(3000,\frac{\ln{6}}{\ln{1.1}}\right)\in f^{-1}$
Yielding $$f^{-1}(3000) = \frac{\ln{6}}{\ln{1.1}}$$

Answer 2

1) Reverse $f(t)$ and $t$

$t = 500(1.1)^{f(t)}$

2) Solve for $f(t)$ (which is really now $f^{-1}(t)$)

Hint: use logarithms

3) Plug in $f^{-1}(3000)$.