Simplify

Problem

ln((x4*(x−4)2)/√(,x2+1))

Solution

1. Apply the quotient rule for logarithms, which states 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 ln(a*b)=ln(a)+ln(b)

ln(x4)+ln((x−4)2)−ln(√(,x2+1))

3. Rewrite the square root as a fractional exponent using the identity √(,u)=u(1/2)

ln(x4)+ln((x−4)2)−ln((x2+1)(1/2))

4. Apply the power rule for logarithms, which states ln(an)=n*ln(a) to each term.

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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