Question: logbase2 3+4logbase2 x + 1/2logbase2 y
1. **State the problem:** Simplify the expression $$\log_2 3 + 4 \log_2 x + \frac{1}{2} \log_2 y$$.
2. **Recall logarithm rules:**
- Power rule: $$a \log_b c = \log_b c^a$$
- Sum rule: $$\log_b m + \log_b n = \log_b (mn)$$
3. **Apply the power rule:**
$$4 \log_2 x = \log_2 x^4$$
$$\frac{1}{2} \log_2 y = \log_2 y^{\frac{1}{2}} = \log_2 \sqrt{y}$$
4. **Rewrite the expression:**
$$\log_2 3 + \log_2 x^4 + \log_2 \sqrt{y}$$
5. **Apply the sum rule:**
$$\log_2 (3 \times x^4 \times \sqrt{y})$$
6. **Final simplified form:**
$$\boxed{\log_2 \left(3 x^4 \sqrt{y}\right)}$$