The amount of the drug is modeled using exponential decay. A number less than 1 times itself, e.g., [tex](0.85)^3=0.61[/tex], gets smaller. A number larger than 1 times itself, e.g., [tex](3.20)^4=104.85[/tex], gets bigger. When the number is less than one, for each successive amount of time, the amount decreases. In this case, it is 0.1 (or 10%) less than 1, which means it decreases by 0.1 times the amount each time. The 20 here is the initial amount.
To find when 12 mg will remain, we need to solve the equation for t. You have [tex]A=20(0.9)^t[/tex], and we want to find when A is equal to 12. So we write [tex]12=20(0.9)^t[/tex]. To isolate t, we first divide each side by 20. [tex] \frac{3}{5} = 0.9^t[/tex]. Then, you take the natural log of each side, so that you have
[tex]ln( \frac{3}{5})=ln(0.9^t) \\ ln(3/5)=t ln(0.9) \\ t = \dfrac{ln(\frac{3}{5})}{ln(\frac{9}{10})} [/tex].