1. **State the problem:** Simplify the expression $$\frac{\log_2 x}{2} - \frac{\log_2 y}{3} - \frac{\log_2 z}{3}$$. 2. **Recall the logarithm power rule:** For any base $a$, $\log_a b^c = c \log_a b$. 3. **Rewrite each term using the power rule in reverse:** $$\frac{\log_2 x}{2} = \log_2 x^{\frac{1}{2}} = \log_2 \sqrt{x}$$ $$\frac{\log_2 y}{3} = \log_2 y^{\frac{1}{3}} = \log_2 \sqrt[3]{y}$$ $$\frac{\log_2 z}{3} = \log_2 z^{\frac{1}{3}} = \log_2 \sqrt[3]{z}$$ 4. **Substitute back into the expression:** $$\log_2 \sqrt{x} - \log_2 \sqrt[3]{y} - \log_2 \sqrt[3]{z}$$ 5. **Use the logarithm subtraction rule:** $\log_a b - \log_a c = \log_a \frac{b}{c}$. 6. **Combine the last two terms:** $$\log_2 \sqrt{x} - \log_2 \left( \sqrt[3]{y} \cdot \sqrt[3]{z} \right) = \log_2 \frac{\sqrt{x}}{\sqrt[3]{y} \cdot \sqrt[3]{z}}$$ 7. **Express the denominator as a single cube root:** $$\sqrt[3]{y} \cdot \sqrt[3]{z} = \sqrt[3]{yz}$$ 8. **Final simplified expression:** $$\boxed{\log_2 \frac{\sqrt{x}}{\sqrt[3]{yz}}}$$ This means the original expression simplifies to the logarithm base 2 of the square root of $x$ divided by the cube root of the product $yz$.