Expand the Logarithmic Expression

Problem

(log_)(√(3,x*z2)/√(,y))

Solution

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

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

2. Rewrite the radicals as fractional exponents using the property √(n,am)=a(m/n)

(log_)((x*z2)(1/3))−(log_)(y(1/2))

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

1/3*(log_)(x*z2)−1/2*(log_)(y)

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

1/3*((log_)(x)+(log_)(z2))−1/2*(log_)(y)

5. Distribute the constant and apply the power rule again to the term (log_)(z2)

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

6. Simplify the coefficients to reach the final expanded form.

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

Final Answer

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

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