1. **State the problem:** Find the value of $\log 3.6$ given $\log 2 = 0.3010$, $\log 3 = 0.4771$, and $\log 5 = 0.6990$. 2. **Recall the logarithm property:** For any positive numbers $a$ and $b$, $\log(ab) = \log a + \log b$. 3. **Express 3.6 as a product of known numbers:** $$3.6 = \frac{36}{10} = \frac{2^2 \times 3^2}{2 \times 5}$$ 4. **Simplify the fraction:** $$\frac{2^2 \times 3^2}{2 \times 5} = \frac{\cancel{2} \times 2 \times 3^2}{\cancel{2} \times 5} = \frac{2 \times 3^2}{5}$$ 5. **Apply the logarithm property:** $$\log 3.6 = \log \left( \frac{2 \times 3^2}{5} \right) = \log 2 + \log 3^2 - \log 5$$ 6. **Use the power rule of logarithms:** $$\log 3^2 = 2 \log 3$$ 7. **Substitute the known values:** $$\log 3.6 = 0.3010 + 2 \times 0.4771 - 0.6990$$ 8. **Calculate:** $$\log 3.6 = 0.3010 + 0.9542 - 0.6990 = 0.5562$$ **Final answer:** $$\boxed{\log 3.6 = 0.5562}$$