Expand the Logarithmic Expression log of ( cube root of xy)/(z^2)

Problem

(log_)(√(3,x*y)/(z2))

Solution

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

(log_)(√(3,x*y)/(z2))=(log_)(√(3,x*y))−(log_)(z2)

2. Rewrite the radical using a fractional exponent, where √(3,A)=A(1/3)

(log_)((x*y)(1/3))−(log_)(z2)

3. Apply the power rule for logarithms, which states that (log_)(Ap)=p*(log_)(A) to both terms.

1/3*(log_)(x*y)−2*(log_)(z)

4. Apply the product rule for logarithms, which states that (log_)(A*B)=(log_)(A)+(log_)(B) to the first term.

1/3*((log_)(x)+(log_)(y))−2*(log_)(z)

5. Distribute the constant to simplify the expression into its final expanded form.

1/3*(log_)(x)+1/3*(log_)(y)−2*(log_)(z)

Final Answer

(log_)(√(3,x*y)/(z2))=1/3*(log_)(x)+1/3*(log_)(y)−2*(log_)(z)

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