Expand the Logarithmic Expression

Problem

ln((x4*(x−4)2)/√(,x2+1))

Solution

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

ln(x4*(x−4)2)−ln(√(,x2+1))

2. Apply the product rule for logarithms to the first term, which states that ln(a*b)=ln(a)+ln(b)

ln(x4)+ln((x−4)2)−ln(√(,x2+1))

3. Rewrite the radical as a fractional exponent using the property √(,a)=a(1/2)

ln(x4)+ln((x−4)2)−ln((x2+1)(1/2))

4. Apply the power rule for logarithms, which states that ln(an)=n*ln(a) to move all exponents to the front of their respective terms.

4*ln(x)+2*ln(x−4)−1/2*ln(x2+1)

Final Answer

ln((x4*(x−4)2)/√(,x2+1))=4*ln(x)+2*ln(x−4)−1/2*ln(x2+1)

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