1. **State the problem:** Evaluate the expression $$\sqrt[3]{\frac{(0.07432)^2 \times 48.38}{8458}}$$ using logarithms. 2. **Recall the logarithm properties:** - $\log(ab) = \log a + \log b$ - $\log\left(\frac{a}{b}\right) = \log a - \log b$ - $\log(a^n) = n \log a$ - To find the cube root, we use the power $\frac{1}{3}$: $\sqrt[3]{x} = x^{\frac{1}{3}}$ 3. **Calculate the logarithm of the numerator:** - First, calculate $\log((0.07432)^2 \times 48.38)$ - Using properties: $\log((0.07432)^2) + \log(48.38) = 2 \log(0.07432) + \log(48.38)$ 4. **Calculate each logarithm:** - $\log(0.07432) \approx -1.129$ - $\log(48.38) \approx 1.685$ 5. **Sum the numerator logs:** - $2 \times (-1.129) + 1.685 = -2.258 + 1.685 = -0.573$ 6. **Calculate the logarithm of the denominator:** - $\log(8458) \approx 3.927$ 7. **Calculate the logarithm of the entire fraction:** - $\log\left(\frac{(0.07432)^2 \times 48.38}{8458}\right) = -0.573 - 3.927 = -4.5$ 8. **Apply the cube root (power $\frac{1}{3}$):** - $\log\left(\sqrt[3]{\frac{(0.07432)^2 \times 48.38}{8458}}\right) = \frac{1}{3} \times (-4.5) = -1.5$ 9. **Find the antilogarithm:** - $10^{-1.5} = 10^{-1} \times 10^{-0.5} = 0.1 \times 0.3162 = 0.03162$ **Final answer:** $$\sqrt[3]{\frac{(0.07432)^2 \times 48.38}{8458}} \approx 0.03162$$