Questions: Write the expression as a logarithm of a single quantity. 2[ln(x) - ln(x + 3) - ln(x - 3)]

Solution Steps

The expression inside the brackets is: \[ \ln(x) - \ln(x+3) - \ln(x-3). \] Using the logarithm subtraction rule, \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\), rewrite the expression as: \[ \ln(x) - \left[\ln(x+3) + \ln(x-3)\right]. \]

The expression \(\ln(x+3) + \ln(x-3)\) can be rewritten using the logarithm addition rule, \(\ln(a) + \ln(b) = \ln(ab)\), as: \[ \ln\left((x+3)(x-3)\right). \] Thus, the expression becomes: \[ \ln(x) - \ln\left((x+3)(x-3)\right). \]

Using the logarithm subtraction rule again, combine the two logarithms: \[ \ln\left(\frac{x}{(x+3)(x-3)}\right). \] Now, multiply the entire expression by 2: \[ 2 \cdot \ln\left(\frac{x}{(x+3)(x-3)}\right). \]

Using the logarithm power rule, \(k \cdot \ln(a) = \ln(a^k)\), rewrite the expression as: \[ \ln\left(\left(\frac{x}{(x+3)(x-3)}\right)^2\right). \]

Final Answer