1. **Problem:** Given the product of $a$ and $\sqrt{b}$ is 31.59, and $\log a = 2.6182$, find $b$ using logarithms to 4 significant figures. 2. **Formula and rules:** - Product rule for logarithms: $\log(ab) = \log a + \log b$ - For square roots: $\sqrt{b} = b^{1/2}$, so $\log \sqrt{b} = \frac{1}{2} \log b$ 3. **Step-by-step solution:** - Given: $a \times \sqrt{b} = 31.59$ - Taking logarithm on both sides: $$\log a + \log \sqrt{b} = \log 31.59$$ - Substitute $\log a = 2.6182$ and $\log \sqrt{b} = \frac{1}{2} \log b$: $$2.6182 + \frac{1}{2} \log b = \log 31.59$$ - Calculate $\log 31.59$: $$\log 31.59 = 1.4997$$ (using common logarithm) - Rearrange to find $\log b$: $$\frac{1}{2} \log b = 1.4997 - 2.6182 = -1.1185$$ $$\log b = 2 \times (-1.1185) = -2.2370$$ - Find $b$ by anti-logarithm: $$b = 10^{-2.2370} = 0.00579$$ (to 4 significant figures) **Final answer:** $b = 0.00579$