Question

select the correct answer
how would you write this expression as a sum of differences?
\\( \log_{2} \left( \sqrt3{x} \cdot y \
ight) \\)

a. \\( \frac{1}{3} \log_{2} x + \log_{2} y \\)
b. \\( \frac{\log_{10} \frac{1}{3}}{\log_{10} 2} \\)
c. \\( \log_{2} x^{3} - \log_{2} y \\)
d. \\( \frac{1}{3} \left( \log_{2} x - \log_{2} y \
ight) \\)

Explanation:

Step1: Recall Logarithm Product Rule

The logarithm of a product is the sum of the logarithms: \(\log_b(MN)=\log_b M+\log_b N\). Also, \(\log_b(M^n)=n\log_b M\) for the power rule. The expression is \(\log_b(\sqrt[3]{x}\cdot y)\), and \(\sqrt[3]{x}=x^{\frac{1}{3}}\).

Step2: Apply Product Rule

Using \(\log_b(MN)=\log_b M+\log_b N\), let \(M = \sqrt[3]{x}=x^{\frac{1}{3}}\) and \(N = y\). So \(\log_b(\sqrt[3]{x}\cdot y)=\log_b(x^{\frac{1}{3}})+\log_b y\).

Step3: Apply Power Rule

Using \(\log_b(M^n)=n\log_b M\) on \(\log_b(x^{\frac{1}{3}})\), we get \(\frac{1}{3}\log_b x+\log_b y\).

Answer:

A. \(\frac{1}{3}\log_{b}x+\log_{b}y\)