Choose the expression that shows an expanded form of the logarithm. \[ \begin{array}{l} \ln \left(\frac{x^{10} \sqrt{x-1}}{3 x-18}\right) \\ 10 \ln x+\left(\frac{1}{2}\right) \ln (x-1)-\ln (3 x-18) \\ \left(\frac{1}{2}\right) \ln x^{10}(x-1)-\ln (3 x-18) \\ 10 \ln x+\ln \sqrt{x-1}-\ln (3 x-12) \end{array} \]
Solution
Step 1 ：Let's start with the first option: \(10 \ln x+\left(\frac{1}{2}\right) \ln (x-1)-\ln (3 x-18)\). The first term \(10 \ln x\) corresponds to \(\ln(x^{10})\), the second term \(\left(\frac{1}{2}\right) \ln (x-1)\) corresponds to \(\ln(\sqrt{x-1})\), and the third term \(-\ln (3 x-18)\) corresponds to \(-\ln(3x-18)\). So, the first option seems to be the correct expanded form of the given logarithm.
Step 2 ：Let's check the second option: \(\left(\frac{1}{2}\right) \ln x^{10}(x-1)-\ln (3 x-18)\). The first term \(\left(\frac{1}{2}\right) \ln x^{10}(x-1)\) doesn't correspond to any part of the given logarithm. The second term \(-\ln (3 x-18)\) corresponds to \(-\ln(3x-18)\). So, the second option is not the correct expanded form of the given logarithm.
Step 3 ：Let's check the third option: \(10 \ln x+\ln \sqrt{x-1}-\ln (3 x-12)\). The first term \(10 \ln x\) corresponds to \(\ln(x^{10})\), the second term \(\ln \sqrt{x-1}\) corresponds to \(\ln(\sqrt{x-1})\), but the third term \(-\ln (3 x-12)\) doesn't correspond to any part of the given logarithm. So, the third option is not the correct expanded form of the given logarithm.
Step 4 ：The correct expanded form of the given logarithm is \(\boxed{10 \ln x+\left(\frac{1}{2}\right) \ln (x-1)-\ln (3 x-18)}\).
