Expand the Logarithmic Expression

Problem

ln(((x2−x−2)/((x+4)2))(1/3))

Solution

1. Apply the power rule for logarithms, ln(ab)=b*ln(a) to move the exponent 1/3 to the front of the expression.

1/3*ln((x2−x−2)/((x+4)2))

2. Apply the quotient rule for logarithms, ln(a/b)=ln(a)−ln(b) to separate the numerator and the denominator.

1/3*(ln(x2−x−2)−ln((x+4)2))

3. Factor the quadratic expression x2−x−2 inside the first logarithm into (x−2)*(x+1)

1/3*(ln((x−2)*(x+1))−ln((x+4)2))

4. Apply the product rule for logarithms, ln(a*b)=ln(a)+ln(b) to the first term and the power rule to the second term.

1/3*(ln(x−2)+ln(x+1)−2*ln(x+4))

5. Distribute the constant 1/3 to each term in the expression.

1/3*ln(x−2)+1/3*ln(x+1)−2/3*ln(x+4)

Final Answer

ln(((x2−x−2)/((x+4)2))(1/3))=1/3*ln(x−2)+1/3*ln(x+1)−2/3*ln(x+4)

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