Question

14 expand the following: \\( \log_2 (xy^2) \\)
a \\( \log_2 x + 2 \log_2 y \\)
b \\( \log_2 x - 2 \log_2 y \\)
c \\( \log_2 x + 2 \log_6 y \\)
d \\( 2 \log_2 x + 2 \log_2 y \\)

Explanation:

Step1: Recall Logarithm Product Rule

The product rule of logarithms states that \(\log_b(MN)=\log_b M+\log_b N\), where \(b>0,b
eq1,M>0,N>0\). For \(\log_2(xy^2)\), we can apply this rule to split the logarithm of the product into the sum of logarithms: \(\log_2(xy^2)=\log_2 x+\log_2 y^2\).

Step2: Recall Logarithm Power Rule

The power rule of logarithms states that \(\log_b(M^n)=n\log_b M\), where \(b>0,b
eq1,M>0,n\in\mathbb{R}\). Applying this rule to \(\log_2 y^2\), we get \(2\log_2 y\).

Step3: Combine the Results

Substituting the result from Step 2 back into the expression from Step 1, we have \(\log_2(xy^2)=\log_2 x + 2\log_2 y\).

Answer:

A. \(\log_{2}x + 2\log_{2}y\)