Given that $\log_{a} y = 2\log_{a} 7 + \log_{a} 4 + \frac{1}{2}$, find $y$ in terms of $a$ - AQA - A-Level Maths Mechanics - Question 7 - 2018 - Paper 3

The error in the student's solution stems from accepting $\frac{3}{2}$ as a valid solution without considering the domain of the logarithm. Logarithmic functions require positive inputs, so $x$ must be greater than zero. Since $x$ cannot be $\frac{-3}{2}$, this solution should be rejected. The solution $\frac{3}{2}$ is acceptable, but the student's context misrepresents that as a valid output. Therefore, the correct interpretation is that $\frac{3}{2}$ is a potential solution, but the negative value must be rejected due to the logarithmic constraints.