jmc

If you call the number of bounces $b$, then this problem can be phrased as: what is the minimum $b$, such that $243\cdot {\left(\frac{2}{3}\right)}^b<30\to {\left(\frac{2}{3}\right)}^b<\frac{30}{243}$, at which point you can take the logarithm of both sides and solve for it exactly, but given that you can use your calculator, it's much easier to just multiply $\frac{2}{3}$ with itself repeatedly until the product is less than $\frac{30}{243}$. Regardless of how you do it, you will get $b=\boxed{6}$.