Expand the Logarithmic Expression log of fifth root of x/y

Problem

(log_)(√(5,x/y))

Solution

1. Rewrite the radical expression using a fractional exponent.

(log_)((x/y)(1/5))

2. Apply the power rule for logarithms, (log_)(an)=n*(log_)(a) to move the exponent to the front.

1/5*(log_)(x/y)

3. Apply the quotient rule for logarithms, (log_)(a/b)=(log_)(a)−(log_)(b) to expand the term inside the parentheses.

1/5*((log_)(x)−(log_)(y))

4. Distribute the fraction to each term to reach the final expanded form.

1/5*(log_)(x)−1/5*(log_)(y)

Final Answer

(log_)(√(5,x/y))=1/5*(log_)(x)−1/5*(log_)(y)

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