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y=ln ((x-2)/(x+2))

y=ln⁡x−2x+2 y=\ln \frac{x-2}{x+2}

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Full solution

Q. y=ln⁡x−2x+2 y=\ln \frac{x-2}{x+2}
  1. Identify Expression: Identify the expression inside the logarithm.\newlineCalculation: (x−2)/(x+2)(x-2)/(x+2)
  2. Apply Logarithmic Property: Apply the logarithmic property: ln⁡(ab)=ln⁡(a)−ln⁡(b)\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b).\newlineCalculation: ln⁡(x−2)−ln⁡(x+2)\ln(x-2) - \ln(x+2)
  3. Check Domain Restrictions: Check for domain restrictions to avoid math errors.\newlineCalculation: x−2x-2 and x+2x+2 must not be zero or negative for the logarithm to be defined.\newlinex−2>0→x>2x-2 > 0 \rightarrow x > 2\newlinex+2>0→x>−2x+2 > 0 \rightarrow x > -2\newlineCombined domain: x>2x > 2

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