Simplify/Condense

Problem

1/3*(2*ln(x+3)+ln(x)−ln(x2−1))

Solution

1. Apply the power property of logarithms to the first term, which states n*ln(a)=ln(an)

1/3*(ln((x+3)2)+ln(x)−ln(x2−1))

2. Apply the product property of logarithms to the first two terms inside the parentheses, which states ln(a)+ln(b)=ln(a*b)

1/3*(ln(x*(x+3)2)−ln(x2−1))

3. Apply the quotient property of logarithms to the terms inside the parentheses, which states ln(a)−ln(b)=ln(a/b)

1/3*ln((x*(x+3)2)/(x2−1))

4. Apply the power property again to move the coefficient 1/3 to the exponent of the argument.

ln(((x*(x+3)2)/(x2−1))1/3)

5. Rewrite the exponent as a cube root to simplify the expression.

ln(√(3,(x*(x+3)2)/(x2−1)))

Final Answer

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

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