The level of radioactivity on the site of a nuclear explosion is decaying exponentially. The level measured in 1970 was found to be 0.7 times the level measured in 1960. What is the half-life. (the amount of years)
Radioactivity in 1970 = $\displaystyle x$
Radioactivity in 1960 = $\displaystyle y$
But $\displaystyle x = 0,7y$
$\displaystyle 0,7y = y(1 - \frac{r}{100} ) ^{10}$
$\displaystyle 0,7 = (1 - \frac{r}{100} ) ^{10}$ [Divide by y]
$\displaystyle (1 - \frac{r}{100} ) = \sqrt[10]{0,7}$
$\displaystyle (1 - \frac{r}{100} ) = 0,9649$
$\displaystyle \frac{r}{100} = 0,0350$
$\displaystyle r = 3,503$
I think this is the rate at which it is decaying. I'll spend a little more time on it...
EDIT: Okay i got it! We now have the rate, so we want to know when $\displaystyle x = 0,5y$
$\displaystyle 0,5y = y(1 - \frac{3,503}{100}) ^ {n} $
$\displaystyle 0,5 = (1 - \frac{3,503}{100}) ^ {n} $
$\displaystyle n = log_{ 1 - \frac{3,503}{100} }{0,5} $
$\displaystyle n = 19,43 \ years $