Evaluate log of eighth root of x/z

Problem

(log_)(√(8,x/z))

Solution

1. Rewrite the radical expression using a fractional exponent.

√(8,x/z)=(x/z)1/8

2. Apply the power rule for logarithms, which states (log_)(ab)=b*(log_)(a)

(log_)((x/z)1/8)=1/8*(log_)(x/z)

3. Apply the quotient rule for logarithms, which states (log_)(a/b)=(log_)(a)−(log_)(b)

1/8*(log_)(x/z)=1/8*((log_)(x)−(log_)(z))

4. Distribute the constant factor to each term in the expression.

1/8*(log_)(x)−1/8*(log_)(z)

Final Answer

(log_)(√(8,x/z))=1/8*(log_)(x)−1/8*(log_)(z)

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