Expand the Logarithmic Expression

Problem

ln((a2*b3)/√(,c))

Solution

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

ln((a2*b3)/√(,c))=ln(a2*b3)−ln(√(,c))

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

ln(a2*b3)−ln(√(,c))=ln(a2)+ln(b3)−ln(√(,c))

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

ln(a2)+ln(b3)−ln(c(1/2))

4. Apply the power rule for logarithms, which states that ln(xp)=p*ln(x) to each term.

2*ln(a)+3*ln(b)−1/2*ln(c)

Final Answer

ln((a2*b3)/√(,c))=2*ln(a)+3*ln(b)−1/2*ln(c)

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