1. **State the problem:** Evaluate the expression $$\frac{(438)^3 \times \sqrt{0.056}}{(388)^4}$$ using logarithms. 2. **Recall the logarithm properties:** - $\log(a \times b) = \log a + \log b$ - $\log\left(\frac{a}{b}\right) = \log a - \log b$ - $\log(a^n) = n \log a$ - $\sqrt{x} = x^{\frac{1}{2}}$ 3. **Rewrite the expression using logarithms:** Let $E = \frac{(438)^3 \times (0.056)^{\frac{1}{2}}}{(388)^4}$. Then, $$\log E = \log(438^3) + \log((0.056)^{\frac{1}{2}}) - \log(388^4)$$ 4. **Apply the power rule:** $$\log E = 3 \log 438 + \frac{1}{2} \log 0.056 - 4 \log 388$$ 5. **Calculate each logarithm (base 10):** - $\log 438 \approx 2.6415$ - $\log 0.056 \approx -1.2518$ - $\log 388 \approx 2.5884$ 6. **Substitute values:** $$\log E = 3 \times 2.6415 + \frac{1}{2} \times (-1.2518) - 4 \times 2.5884$$ $$= 7.9245 - 0.6259 - 10.3536$$ $$= -3.0550$$ 7. **Find the value of $E$ by taking the antilog:** $$E = 10^{-3.0550} \approx 0.00088$$ **Final answer:** $$\boxed{0.00088}$$