Calculate how long it take to reach a goal

Question

Given a growth rate for a period and a goal. How can I calculate how many periods that it will take to reach that goal.

Example:

An investment currently valued at $400 grows at 30% per week.

With a goal of $2000, how can I calculate how many periods(weeks) that it will take to reach that goal?

Answer 1

Each week you multiply the principal by $1+30\%=1.3$ After $w$ weeks, you would have $400(1.3)^w$ You want $2000=400(1.3^w)$ or $5=1.3^w$. Now if you take the logarithm,

$\log 5 = w \log 1.3 \\ w=\frac {\log 5}{\log 1.3}$

You can use any base of logs you like.