Questions: Expand: [ ln left[fracsqrt[3]x-1(3 x-2)^4(x+1) sqrtx-1right]^2 2 ]

Solution Steps

To expand the given logarithmic expression, we will use the properties of logarithms. Specifically, we will apply the power rule, product rule, and quotient rule for logarithms. The power rule allows us to bring the exponent in front of the logarithm. The quotient rule allows us to express the logarithm of a quotient as the difference of two logarithms, and the product rule allows us to express the logarithm of a product as the sum of two logarithms.

We start with the expression given in the problem: \[ \ln \left[\frac{\sqrt[3]{x-1}(3 x-2)^{4}}{(x+1) \sqrt{x-1}}\right]^{2} \] This can be rewritten as: \[ \ln \left[\frac{(3x - 2)^{4}}{(x + 1) \sqrt{x - 1}} \cdot \frac{1}{\sqrt[3]{x - 1}}\right]^{2} \]

We simplify the expression inside the logarithm: \[ \frac{(3x - 2)^{4}}{(x + 1) \sqrt{x - 1} \cdot \sqrt[3]{x - 1}} = \frac{(3x - 2)^{4}}{(x + 1)(x - 1)^{\frac{5}{6}}} \] Thus, the expression becomes: \[ \ln \left[\frac{(3x - 2)^{4}}{(x + 1)(x - 1)^{\frac{5}{6}}}\right]^{2} \]

Using the power rule of logarithms, we can bring the exponent down: \[ 2 \ln \left[\frac{(3x - 2)^{4}}{(x + 1)(x - 1)^{\frac{5}{6}}}\right] \] Next, we apply the quotient rule: \[ 2 \left( \ln((3x - 2)^{4}) - \ln((x + 1)(x - 1)^{\frac{5}{6}}) \right) \] This further simplifies to: \[ 2 \left( 4 \ln(3x - 2) - \left( \ln(x + 1) + \frac{5}{6} \ln(x - 1) \right) \right) \]

Final Answer