1. **State the problem:** We need to evaluate the expression $$\sqrt{\frac{3.416 \times 0.0789}{\frac{691.6 \times 1.41}{1000}}}$$ using logarithm tables. 2. **Recall the logarithm properties:** - $\log(ab) = \log a + \log b$ - $\log\left(\frac{a}{b}\right) = \log a - \log b$ - $\log(\sqrt{x}) = \frac{1}{2} \log x$ 3. **Calculate the numerator:** - $3.416 \times 0.0789$ - Using logs: $\log 3.416 + \log 0.0789$ 4. **Calculate the denominator:** - $\frac{691.6 \times 1.41}{1000} = \frac{691.6 \times 1.41}{1000}$ - Using logs: $\log 691.6 + \log 1.41 - \log 1000$ 5. **Find logarithms from tables:** - $\log 3.416 \approx 0.5335$ - $\log 0.0789 \approx -1.1038$ - $\log 691.6 \approx 2.8401$ - $\log 1.41 \approx 0.1492$ - $\log 1000 = 3$ 6. **Calculate numerator log:** $$0.5335 + (-1.1038) = -0.5703$$ 7. **Calculate denominator log:** $$2.8401 + 0.1492 - 3 = -0.0107$$ 8. **Calculate the log of the fraction:** $$-0.5703 - (-0.0107) = -0.5596$$ 9. **Calculate the log of the square root:** $$\frac{1}{2} \times (-0.5596) = -0.2798$$ 10. **Find the antilog:** - $\text{Antilog}(-0.2798) = 10^{-0.2798} \approx 0.527$ **Final answer:** $$\sqrt{\frac{3.416 \times 0.0789}{\frac{691.6 \times 1.41}{1000}}} \approx 0.527$$