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$\log_{2}\left(\frac{x^{3}}{16}\right)$

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Quotient property of logarithms

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Q.

\log_{2}\left(\frac{x^{3}}{16}\right)

Apply Quotient Rule: Apply the quotient rule of logarithms to $\log_2\left(\frac{x^3}{16}\right)$ . The quotient rule of logarithms states that $\log_b\left(\frac{a}{c}\right) = \log_b(a) - \log_b(c)$ . We can apply this rule to the given logarithm. $\log_2\left(\frac{x^3}{16}\right) = \log_2(x^3) - \log_2(16)$
Simplify Log $16$ : Simplify $\log_2(16)$ using the fact that $16$ is a power of $2$ . $\newline$ Since $16$ is $2$ to the power of $4$ , we can write $\log_2(16)$ as $\log_2(2^4)$ . According to the power rule of logarithms, $\log_b(a^n) = n \cdot \log_b(a)$ , so $\log_2(2^4) = 4 \cdot \log_2(2)$ . Since $16$ $0$ is $16$ $1$ , $\log_2(16)$ simplifies to $4$ . $\newline$ $16$ $4$
Apply Power Rule: Apply the power rule of logarithms to $\log_2(x^3)$ . The power rule of logarithms states that $\log_b(a^n) = n \cdot \log_b(a)$ . We can apply this rule to $\log_2(x^3)$ . $\log_2(x^3) = 3 \cdot \log_2(x)$
Combine Results: Combine the results from Step $2$ and Step $3$ . $\newline$ We have $\log_2(x^3)$ as $3 \cdot \log_2(x)$ and $\log_2(16)$ as $4$ . Now we combine these results to get the final expanded form of the original logarithm. $\newline$ $\log_2\left(\frac{x^3}{16}\right) = 3 \cdot \log_2(x) - 4$

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