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I start with say 5. multiplied by 1.5 per second. How do I calculate how many seconds I need to reach say 7000?

(im trying to make a formula for a table top adventure I'm planning for the weekend)

For now im using spreadsheets with tables per second for this. Would be awesome to be able to calculate it with a formula.

Rachael Zacc
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  • Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or closed. To prevent that, please [edit] the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Apr 01 '22 at 07:43
  • Hint: population after 2 seconds: $N=5\times 1.5^2$. Set this for n seconds and use logarithms to ease the process :) – ck1987pd Apr 01 '22 at 07:48
  • @C.Koca learning about logarithms now. – Rachael Zacc Apr 01 '22 at 07:54

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You start with 5. After one second, you get $5*1.5=7,5$. After two second you get $7.5*1.5=5*1.5*1.5=5*(1.5)^2=11.25$ ... After n seconds, you get $5*(1.5)^n$

That is, let $s_{start}$ be your starting number and $s_{final}$ your final number, $\epsilon$ the number you multiply by, and $n$ the number of iterations. Then $$ s_{final}=s_{start}*\epsilon^n $$

Therefore, $$ \frac{s_{final}}{s_{start}}=\epsilon^n $$ and finaly, $$ log_\epsilon(\frac{s_{final}}{s_{start}})=n $$ To answer your example, $s_{final}=7000$, $s_{start}=5$, $\epsilon=1.5$, then $$ n=17.8665 $$

Baloo
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