Simplify/Condense natural log of x^2-4- natural log of x+2

Problem

ln(x2−4)−ln(x+2)

Solution

1. Identify the logarithmic property for subtraction, which states that the difference of two logarithms with the same base is the logarithm of the quotient of their arguments.

ln(a)−ln(b)=ln(a/b)

2. Apply the quotient rule to combine the two natural logarithms into a single expression.

ln((x2−4)/(x+2))

3. Factor the numerator using the difference of squares formula, where x2−4 becomes (x−2)*(x+2)

ln(((x−2)*(x+2))/(x+2))

4. Simplify the expression by canceling the common factor of (x+2) from the numerator and the denominator, assuming x≠−2

ln(x−2)

Final Answer

ln(x2−4)−ln(x+2)=ln(x−2)

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