Choose the expression that shows an expanded form of the logarithm. \[ \begin{array}{l}\ln \left(\frac{x^{20} \sqrt{x-1}}{3 x-11}\right) \\ 20 \ln x+\ln \sqrt{x-1}-\ln (3 x-12) \\ \left(\frac{1}{2}\right) \ln x^{20}(x-1)-\ln (3 x-11) \\ 20 \ln x+\left(\frac{1}{2}\right) \ln (x-1)-\ln (3 x-11)\end{array} \]

The expression that shows an expanded form of the logarithm is: \[ 20 \ln x + \left(\frac{1}{2}\right) \ln (x-1) - \ln (3x-11) \] This expression has successfully applied the properties of logarithms, where: 1. The logarithm of a quotient is the difference of the logarithms. 2. The logarithm of a product is the sum of the logarithms. 3. Additionally, the logarithm of a power can be expressed as a multiplication of the exponent by the logarithm of the base. This makes it a great representation of logarithmic expansion! Now, let's talk about common pitfalls with logarithmic expressions. One common mistake students make is forgetting to apply the properties correctly, like mixing up addition and subtraction when moving between logarithmic forms. Always ensure you're clear about whether you're dealing with a sum (product rule) or a difference (quotient rule) to avoid miscalculations!