Expand the Logarithmic Expression log of (4x^3)/(y^2(x-1)^5)

Problem

(log_)((4*x3)/(y2*(x−1)5))

Solution

1. Apply the quotient rule for logarithms, which states that (log_)(A/B)=(log_)(A)−(log_)(B)

(log_)(4*x3)−(log_)(y2*(x−1)5)

2. Apply the product rule for logarithms to both terms, using (log_)(A*B)=(log_)(A)+(log_)(B)

(log_)(4)+(log_)(x3)−((log_)(y2)+(log_)((x−1)5))

3. Distribute the negative sign through the second part of the expression.

(log_)(4)+(log_)(x3)−(log_)(y2)−(log_)((x−1)5)

4. Apply the power rule for logarithms, which states that (log_)(An)=n*(log_)(A) to move the exponents to the front of each term.

(log_)(4)+3*(log_)(x)−2*(log_)(y)−5*(log_)(x−1)

Final Answer

(log_)((4*x3)/(y2*(x−1)5))=(log_)(4)+3*(log_)(x)−2*(log_)(y)−5*(log_)(x−1)

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