Questions: Use the properties of logarithms to condense the following expression. [ (1/2) ln x+ln (x^2-1)-ln (x+1) ]

Solution Steps

The expression contains \(\frac{1}{2} \ln x\). Using the power rule of logarithms, \(\ln a^b = b \ln a\), we can rewrite this term as: \[ \frac{1}{2} \ln x = \ln x^{\frac{1}{2}} = \ln \sqrt{x}. \]

The expression now is: \[ \ln \sqrt{x} + \ln (x^2 - 1). \] Using the product rule of logarithms, \(\ln a + \ln b = \ln (a \cdot b)\), we combine these terms: \[ \ln \sqrt{x} + \ln (x^2 - 1) = \ln \left( \sqrt{x} \cdot (x^2 - 1) \right). \]

The expression now is: \[ \ln \left( \sqrt{x} \cdot (x^2 - 1) \right) - \ln (x + 1). \] Using the quotient rule of logarithms, \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\), we combine these terms: \[ \ln \left( \sqrt{x} \cdot (x^2 - 1) \right) - \ln (x + 1) = \ln \left( \frac{\sqrt{x} \cdot (x^2 - 1)}{x + 1} \right). \]

Final Answer