\begin{tabular}{l} Write the expression as a single logarithm. \ $$ \qquad 3\left(\log _{3} z-5 \log _{3} y ight)+3 \log _{3} w $$ \ \hline lognt (D) \end{tabular}

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\begin{tabular}{l} Write the expression as a single logarithm. \ $$ \qquad 3\left(\log _{3} z-5 \log _{3} yight)+3 \log _{3} w $$ \ \hline lognt (D) \end{tabular}

UpStudy AI · Accepted Answer

To combine the expression $$ 3\left(\log _{3} z-5 \log _{3} y\right)+3 \log _{3} w $$ into a single logarithm, we will follow these steps:

1. Distribute the $$3$$ in the first part of the expression.
2. Combine the logarithmic terms using the properties of logarithms.

Let's start with the distribution:

$$
3\left(\log _{3} z - 5 \log _{3} y\right) = 3 \log _{3} z - 15 \log _{3} y
$$

Now, we can rewrite the entire expression:

$$
3 \log _{3} z - 15 \log _{3} y + 3 \log _{3} w
$$

Next, we can factor out the $$3$$ from the logarithmic terms:

$$
3\left(\log _{3} z + \log _{3} w - 15 \log _{3} y\right)
$$

Using the property of logarithms that states $$a \log_b c = \log_b(c^a)$$, we can rewrite $$ -15 \log_3 y $$ as $$ \log_3(y^{-15}) $$:

$$
3\left(\log _{3} z + \log _{3} w + \log _{3}(y^{-15})\right)
$$

Now, we can combine the logarithms inside the parentheses using the property that states $$\log_b a + \log_b c = \log_b(ac)$$:

$$
3\left(\log _{3}(z \cdot w \cdot y^{-15})\right)
$$

Finally, we can apply the factor of $$3$$ outside the logarithm:

$$
\log_3((z \cdot w \cdot y^{-15})^3)
$$

Thus, the expression as a single logarithm is:

$$
\log_3\left(\frac{(z \cdot w)^3}{y^{15}}\right)
$$

This is the final answer.
The expression simplifies to $$ \log_3\left(\frac{(z \cdot w)^3}{y^{15}}\right) $$.

\begin{tabular}{l} Write the expression as a single logarithm. \\ \( \qquad 3\left(\log _{3} z-5 \log _{3} y\right)+3 \log _{3} w \) \\ \hline lognt (D) \end{tabular}

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