2025 MSHS PRELIMS P2 Q5

Part (a): the division sign is restored from the 2025 Maris Stella paper. For the requested single-power form, assume \(y>0\); this sign condition is an editorial clarification.

1. Find the integer \(a\) and \(b\) which satisfy the equation \[ \left(\frac{1}{8}y^3\right)^{-1}\div(4y^2)^{\frac{1}{2}}=2^a y^b. \][3]
2. Solve the equation \(\log_3(x-8)=2-\frac{1}{\log_x 3}\).[5]
3. Solve the equation \(4^{x+1}+7(2^x)=2\).[4]