Questions: ln(x/(x-9))+ln((x+9)/x)-ln(x^2-81) ln(x/(x-9))+ln((x+9)/x)-ln(x^2-81)=□

Solution Steps

To combine the given logarithmic expression into a single logarithm, we can use the properties of logarithms. Specifically, we will use the product rule \(\ln(a) + \ln(b) = \ln(a \cdot b)\) and the quotient rule \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\). First, combine the first two logarithms using the product rule, then apply the quotient rule to combine the result with the third logarithm.

The given expression is: \[ \ln\left(\frac{x}{x-9}\right) + \ln\left(\frac{x+9}{x}\right) - \ln\left(x^2 - 81\right) \] First, apply the product rule of logarithms to combine the first two terms: \[ \ln\left(\frac{x}{x-9}\right) + \ln\left(\frac{x+9}{x}\right) = \ln\left(\frac{x}{x-9} \cdot \frac{x+9}{x}\right) \]

Simplify the expression inside the logarithm: \[ \frac{x}{x-9} \cdot \frac{x+9}{x} = \frac{x(x+9)}{x(x-9)} = \frac{x+9}{x-9} \]

Now, apply the quotient rule to combine the result with the third logarithm: \[ \ln\left(\frac{x+9}{x-9}\right) - \ln\left(x^2 - 81\right) = \ln\left(\frac{x+9}{x-9} \cdot \frac{1}{x^2 - 81}\right) \]

Recognize that \(x^2 - 81\) can be factored as \((x-9)(x+9)\): \[ \ln\left(\frac{x+9}{x-9} \cdot \frac{1}{(x-9)(x+9)}\right) = \ln\left(\frac{1}{x-9}\right) \]

Final Answer