Question 18

Explanation

Let $a = \log_3 x$ and $b = \log_2 y$. The first equation will be obtained

Form $\log_3 x \cdot \log_2 y = a \cdot b$

This means the maximum value of $\log_3 x \cdot \log_2 y$ is equal to the maximum value of the function $f(b) = 5b - b^2$. The condition for the maximum of the function is $f'(b) = 0$

Thus the value of $a$

So the value of $ab$ will be maximum when $a = \frac{5}{2}$ and $b = \frac{5}{2}$. The result of their product is