Write the expression in terms of \( \ln x, \ln (x+4) \), and \( \ln (x+5) \) \( \ln \frac{(x+4)^{5}}{x^{5} \sqrt{x+5}}=\square \) \( \ln \left(\sqrt{x+4}\left(\frac{x}{x+5}\right)^{7}\right)=\square \)

To rewrite the expression \( \ln \frac{(x+4)^{5}}{x^{5} \sqrt{x+5}} \) in terms of \( \ln x, \ln (x+4) \), and \( \ln (x+5) \), we can use the properties of logarithms. 1. Applying the quotient rule for logarithms, we can break it down: \( \ln \frac{(x+4)^{5}}{x^{5} \sqrt{x+5}} = \ln (x+4)^{5} - \ln (x^{5} \sqrt{x+5}) \). 2. Now we simplify each term: - \( \ln (x+4)^{5} = 5 \ln (x+4) \). - \( \ln (x^{5}) = 5 \ln x \). - \( \ln (\sqrt{x+5}) = \frac{1}{2} \ln (x+5) \). 3. Putting this together gives us: \[ 5 \ln (x+4) - (5 \ln x + \frac{1}{2} \ln (x+5)) = 5 \ln (x+4) - 5 \ln x - \frac{1}{2} \ln (x+5). \] So, the filled square is: \[ 5 \ln (x + 4) - 5 \ln x - \frac{1}{2} \ln (x + 5). \] For the second expression \( \ln \left(\sqrt{x+4} \left(\frac{x}{x+5}\right)^{7}\right) \): 1. Again using properties of logarithms, we can break it down: \( \ln \left(\sqrt{x+4} \left(\frac{x}{x+5}\right)^{7}\right) = \ln \sqrt{x+4} + \ln \left(\frac{x}{x+5}\right)^{7} \). 2. Now, simplifying each term: - \( \ln \sqrt{x+4} = \frac{1}{2} \ln (x+4) \). - \( \ln \left(\frac{x}{x+5}\right)^{7} = 7 \ln \left(\frac{x}{x+5}\right) = 7 (\ln x - \ln (x+5)) \). 3. Putting this together gives us: \[ \frac{1}{2} \ln (x+4) + 7 \ln x - 7 \ln (x+5). \] So, the filled square for this expression is: \[ \frac{1}{2} \ln (x + 4) + 7 \ln x - 7 \ln (x + 5). \]