i finished about all of this math problem except for the ending.
i reduced the problem to 5000^-10k
what i need is how to find the value of k
we have $\displaystyle 450 = 500 \cdot 10^{-10k}$
$\displaystyle \Rightarrow \frac {450}{500} = 10^{-10k}$
$\displaystyle \Rightarrow \log \left( \frac {9}{10} \right) = -10k$ ........do you understand this step?
$\displaystyle \Rightarrow k = \frac {\log \left( \frac {9}{10} \right)}{-10}$