Expand the Logarithmic Expression log of (a^2)/(b^4 square root of c)

Problem

(log_)((a2)/(b4√(,c)))

Solution

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

(log_)((a2)/(b4√(,c)))=(log_)(a2)−(log_)(b4√(,c))

2. Apply the product rule for logarithms to the second term, which states that (log_)(M*N)=(log_)(M)+(log_)(N)

(log_)(a2)−((log_)(b4)+(log_)(√(,c)))

3. Distribute the negative sign through the parentheses.

(log_)(a2)−(log_)(b4)−(log_)(√(,c))

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

(log_)(a2)−(log_)(b4)−(log_)(c(1/2))

5. Apply the power rule for logarithms, which states that (log_)(Mp)=p*(log_)(M) to each term.

2*(log_)(a)−4*(log_)(b)−1/2*(log_)(c)

Final Answer

(log_)((a2)/(b4√(,c)))=2*(log_)(a)−4*(log_)(b)−1/2*(log_)(c)

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