Question

1. apply the product property, quotient property, and/or power property to expand the logarithmic expression \\(\log_{5} 6x^{2}\\)

\\(\bigcirc\\) \\(3\log_{5}3 + 3\log_{5}2 + 2\log_{5}x\\)
\\(\bigcirc\\) \\(\log_{5}5 + \log_{5}6 + 2\log_{5}x\\)
\\(\bigcirc\\) \\(\log_{5}2 + \log_{5}4 + 2\log_{5}x\\)
\\(\bigcirc\\) \\(\log_{5}3 + \log_{5}2 + 2\log_{5}x\\)

Explanation:

Step1: Apply Product Property

The product property of logarithms states that $\log_b(MN) = \log_b M + \log_b N$. For $\log_5(6x^2)$, we can split $6x^2$ into $6 \times x^2$, so $\log_5(6x^2)=\log_5 6 + \log_5 x^2$.

Step2: Apply Power Property

The power property of logarithms states that $\log_b(M^n)=n\log_b M$. For $\log_5 x^2$, this becomes $2\log_5 x$. Now we have $\log_5 6 + 2\log_5 x$.

Step3: Factor 6

Since $6 = 3\times2$, we can apply the product property again to $\log_5 6$. So $\log_5 6=\log_5(3\times2)=\log_5 3 + \log_5 2$. Combining this with the previous result, we get $\log_5 3 + \log_5 2 + 2\log_5 x$.

Answer:

$\log_{5}3 + \log_{5}2 + 2\log_{5}x$ (corresponding to the option: $\boldsymbol{\log_{5}3 + \log_{5}2 + 2\log_{5}x}$)