Evaluate log base 9 of (19 square root of x)/y

Problem

(log_9)((19√(,x))/y)

Solution

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

(log_9)(19√(,x))−(log_9)(y)

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

(log_9)(19)+(log_9)(√(,x))−(log_9)(y)

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

(log_9)(19)+(log_9)(x(1/2))−(log_9)(y)

4. Apply the power rule for logarithms, which states that (log_b)(Mp)=p*(log_b)(M) to the middle term.

(log_9)(19)+1/2*(log_9)(x)−(log_9)(y)

Final Answer

(log_9)((19√(,x))/y)=(log_9)(19)+1/2*(log_9)(x)−(log_9)(y)

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