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I multiply 100 by 1.05. I get 105. I multiply 105 by 1.05. I get 110.25. I multiply 110.25 by 1.05. I get 115,7625 and so on.

If I choose a fully random non-natural positive rational number, for example 240.353 What's the formula to determine how much I need to multiply 100 by 1.05 to get this number?

The only thing I know is that the number of times I have to multiply 1.05 to get any random non-natural number is also going to be non-natural.

Codem
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  • Have you learned logarithms yet? – JimmyK4542 Nov 09 '20 at 03:47
  • For your rational number $r$, you want to find out the value $k$ for which $100\cdot 105^k = r$. If $\log(100\cdot 105^k) = \log r$, then $100\cdot 105^k = r$. Can you solve the equation with $\log$ in it for $k$? – Steve Kass Nov 09 '20 at 03:47

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Given an arbitrary positive rational number $y$, you are looking for a real number $x$ such that $$100 (1.05)^x =y $$ $$ 1.05 ^x = y/(100) $$

$$x= Ln ( y/100) /Ln (1.05)$$

Mohammad Riazi-Kermani
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