Question

ii. apply the product property, quotient property, and/or power property to expand the logarithmic expression \\(\log_{2} \frac{32}{5} t^{6}\\).\
\\(\bigcirc\\) \\(\log_{2} 6.5 + 6\log_{2} t\\)\
\\(\bigcirc\\) \\(5 - \log_{2} 5 + 6\log_{2} t\\)\
\\(\bigcirc\\) \\(6 - \log_{2} 5 + 6\log_{2} t\\)\
\\(\bigcirc\\) \\(5 + \log_{2} 5 + 6\log_{2} t\\)

Explanation:

Step1: Apply Product Property

The given expression is \( \log_{2}\frac{32}{5}f^{6} \). By the product property of logarithms \( \log_b(xy)=\log_b x+\log_b y \), we can split it as \( \log_{2}\frac{32}{5}+\log_{2}f^{6} \).

Step2: Apply Quotient Property

For \( \log_{2}\frac{32}{5} \), using the quotient property \( \log_b\frac{x}{y}=\log_b x - \log_b y \), we get \( \log_{2}32-\log_{2}5 \).

Step3: Apply Power Property

For \( \log_{2}f^{6} \), using the power property \( \log_b x^n = n\log_b x \), we get \( 6\log_{2}f \).

Step4: Simplify \( \log_{2}32 \)

Since \( 2^5 = 32 \), by the definition of logarithms \( \log_b b^x=x \), we have \( \log_{2}32 = 5 \).

Step5: Combine Terms

Substituting back, we have \( 5-\log_{2}5 + 6\log_{2}f \).

Answer:

B. \( 5 - \log_{2}5 + 6\log_{2}f \)