Expand the Logarithmic Expression log base a of (x^6)/(yz^7)

Problem

(log_a)((x6)/(y*z7))

Solution

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

(log_a)(x6)−(log_a)(y*z7)

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

(log_a)(x6)−((log_a)(y)+(log_a)(z7))

3. Distribute the negative sign through the parentheses.

(log_a)(x6)−(log_a)(y)−(log_a)(z7)

4. Apply the power rule for logarithms, which states that (log_b)(Mp)=p*(log_b)(M) to the first and third terms.

6*(log_a)(x)−(log_a)(y)−7*(log_a)(z)

Final Answer

(log_a)((x6)/(y*z7))=6*(log_a)(x)−(log_a)(y)−7*(log_a)(z)

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