1. **State the problem:** Simplify the expression $$\log_m \sqrt{a} + 4 \log_m (m^4 a) - 2$$ into a single logarithm of the form $$\log_m (a^x m^y)$$ and find the values of $x$ and $y$. 2. **Recall logarithm rules:** - $$\log_b (xy) = \log_b x + \log_b y$$ - $$\log_b (x^r) = r \log_b x$$ - $$\log_b b = 1$$ 3. **Rewrite each term:** - $$\log_m \sqrt{a} = \log_m a^{\frac{1}{2}} = \frac{1}{2} \log_m a$$ - $$4 \log_m (m^4 a) = 4 \left( \log_m m^4 + \log_m a \right) = 4 (4 + \log_m a) = 16 + 4 \log_m a$$ - The constant $$-2$$ can be written as $$-2 = -2 \log_m m$$ since $$\log_m m = 1$$. 4. **Combine all terms:** $$\frac{1}{2} \log_m a + 16 + 4 \log_m a - 2 \log_m m = \left( \frac{1}{2} + 4 \right) \log_m a + (16 - 2) \log_m m = \frac{9}{2} \log_m a + 14 \log_m m$$ 5. **Express as a single logarithm:** $$\frac{9}{2} \log_m a + 14 \log_m m = \log_m a^{\frac{9}{2}} + \log_m m^{14} = \log_m \left(a^{\frac{9}{2}} m^{14} \right)$$ 6. **Final answer:** - $$x = \frac{9}{2}$$ - $$y = 14$$