1. **State the problem:** We need to express the expression $3 \log x - \frac{1}{2} \log y + 1$ as a single logarithm. 2. **Rewrite the constant term as a logarithm:** Recall that $1 = \log b$ where $b$ depends on the base of the logarithm. Without loss of generality, assume the logarithm is base 10, so $1 = \log 10$. 3. **Use logarithm power rule:** $a \log b = \log b^a$ to rewrite each term with coefficients. $$3 \log x = \log x^3$$ $$- \frac{1}{2} \log y = \log y^{-\frac{1}{2}} = \log \frac{1}{\sqrt{y}}$$ 4. **Rewrite the expression using these rules:** $$3 \log x - \frac{1}{2} \log y + 1 = \log x^3 + \log \frac{1}{\sqrt{y}} + \log 10$$ 5. **Use logarithm addition rule:** $\log a + \log b = \log(ab)$, so combine all logarithms into one: $$\log \biggl(x^3 \cdot \frac{1}{\sqrt{y}} \cdot 10 \biggr) = \log \frac{10 x^3}{\sqrt{y}}$$ **Final answer:** $$\boxed{\log \frac{10 x^3}{\sqrt{y}}}$$