You are exactly right about what you need to do. If you can't use logarithms, you have to use trial and error, which is greatly helped by a good graph.
In this kind of problem, you may be able to get ONLY an approximate answer. Your graph should suggest a bit less than 90 minutes. Good work with your approximation of 87. The quickest way to get an excellent approximation is to use logarithms as suggested by Denis.
$x = \dfrac{30 * log \left ( \dfrac{8}{60} \right )}{ log \left ( \dfrac{1}{2} \right )} \approx 87.2067.$
Now how do you check an answer?