Simplify/Condense

Problem

ln(x/(x−3))+ln((x+3)/x)−ln(x2−9)

Solution

1. Apply the product rule for logarithms to the first two terms, which states ln(a)+ln(b)=ln(a*b)

ln(x/(x−3)⋅(x+3)/x)−ln(x2−9)

2. Simplify the product inside the first logarithm by canceling the common factor x

ln((x+3)/(x−3))−ln(x2−9)

3. Apply the quotient rule for logarithms, which states ln(a)−ln(b)=ln(a/b)

ln((x+3)/(x−3)/(x2−9))

4. Factor the denominator using the difference of squares formula x2−9=(x−3)*(x+3)

ln((x+3)/(x−3)/((x−3)*(x+3)))

5. Simplify the fraction by multiplying the numerator by the reciprocal of the denominator.

ln((x+3)/(x−3)⋅1/((x−3)*(x+3)))

6. Cancel the common factor (x+3) and combine the remaining terms in the denominator.

ln(1/((x−3)2))

7. Apply the power rule for logarithms, ln(a(−n))=−n*ln(a) to rewrite the expression if desired.

−2*ln(x−3)

Final Answer

ln(x/(x−3))+ln((x+3)/x)−ln(x2−9)=−2*ln(x−3)

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