# Find the number of hours it would take to reach 50. I know that multiplying a previous value by 0.921 gives you the next value for a whole hour (bottom to top). However, I am really struggling to find a way to calculate the hours for the value 50. How can I go about doing this?

Probability - Geometric Sequence Problem
I have been trying to solve a problem which states the following:
For every hour less than 8, an initial value depreciates by 7.9%. The initial value given is 87. This is what I have calculated so far:
8 hours = 87
7 hours = 80.127
6 hours = 73.797
5 hours = 67.967
4 hours = 62.598
3 hours = 57.652
2 hours = 53.098
1 hours = 48.903
From the above, it can be seen that the problem is a geometric sequence, having a constant ratio of 0.921.
The problem asks to find the number of hours it would take to reach 50. I know that multiplying a previous value by 0.921 gives you the next value for a whole hour (bottom to top). However, I am really struggling to find a way to calculate the hours for the value 50. How can I go about doing this?
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yatangije62
Step 1
After each hour, the value V is multiplied by 0.921 (i.e. losing 7.9%).
So after h hours, the value is the initial value, 87, multiplied by 0.921, h times:
$V=87\ast 0.921\ast 0.921\ast ...\ast 0.921$ (h occurences of 0.921)
This can be written as $V=87\ast {0.921}^{h}$, and the formula works for fractional values of h as well as for whole numbers.
Step 2
Rearrange this formula by taking logarithms of both sides (noting that $log\left(A\ast B\right)=log\left(A\right)+log\left(B\right)$ and $log\left({A}^{C}\right)=C\ast log\left(A\right)$
So: $log\left(V\right)=log\left(87\right)+h\ast log\left(0.921\right)$
You can then substitute in the value $V=50$ to solve for h