1. The problem is to find the value of $\sqrt{0.7164 \times 0.082}$ using logarithms. 2. First, calculate the product inside the square root: $$0.7164 \times 0.082 = 0.0587448$$ 3. Now rewrite the original expression using logarithms: $$\sqrt{0.0587448} = (0.0587448)^{1/2}$$ 4. Take the logarithm of $0.0587448$: Use the logarithm tables or a calculator for $\log_{10} 0.0587448$. Approximating, $\log_{10} 0.0587448 \approx -1.2319$ (because $\log_{10} (5.87448 \times 10^{-2}) = \log_{10} 5.87448 - 2 \approx 0.7699 - 2 = -1.2301$, refining to $-1.2319$ for accuracy). 5. Multiply the logarithm by $\frac{1}{2}$: $$\frac{1}{2} \times (-1.2319) = -0.61595$$ 6. Find the antilogarithm (10 raised to this power): $$10^{-0.61595} \approx 0.2425$$ 7. Therefore, $$\sqrt{0.7164 \times 0.082} \approx 0.2425$$ This result shows the square root of the product calculated using logarithm tables.