Expand the Logarithmic Expression natural log of x/( square root of x^2+1)

Problem

ln(x/√(,x2+1))

Solution

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

ln(x/√(,x2+1))=ln(x)−ln(√(,x2+1))

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

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

3. Apply the power rule for logarithms, which states that ln(ap)=p*ln(a) to move the exponent to the front of the term.

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

Final Answer

ln(x/√(,x2+1))=ln(x)−1/2*ln(x2+1)

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