$$ 1 \leftarrow \quad \begin{array}{l} ext { Rewrite as sums or differences of logarithms. } \ \log _{b}\left(x^{3} y^{5} z ight) \ \log _{b}\left(x^{3} y^{5} z ight)=\square\end{array} $$

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$$ 1 \leftarrow \quad \begin{array}{l}	ext { Rewrite as sums or differences of logarithms. } \ \log _{b}\left(x^{3} y^{5} zight) \ \log _{b}\left(x^{3} y^{5} zight)=\square\end{array} $$

UpStudy AI · Accepted Answer

To rewrite the expression $$\log_{b}(x^{3} y^{5} z)$$ as sums or differences of logarithms, we can use the properties of logarithms:

1. The logarithm of a product: $$\log_{b}(mn) = \log_{b}(m) + \log_{b}(n)$$
2. The logarithm of a power: $$\log_{b}(m^n) = n \cdot \log_{b}(m)$$

Using these properties, we can break down $$\log_{b}(x^{3} y^{5} z)$$ as follows:

$$
\log_{b}(x^{3} y^{5} z) = \log_{b}(x^{3}) + \log_{b}(y^{5}) + \log_{b}(z)
$$

Now, applying the power property to the first two terms:

$$
\log_{b}(x^{3}) = 3 \cdot \log_{b}(x)
$$
$$
\log_{b}(y^{5}) = 5 \cdot \log_{b}(y)
$$

Putting it all together, we have:

$$
\log_{b}(x^{3} y^{5} z) = 3 \cdot \log_{b}(x) + 5 \cdot \log_{b}(y) + \log_{b}(z)
$$

Thus, the final expression is:

$$
\log_{b}(x^{3} y^{5} z) = 3 \log_{b}(x) + 5 \log_{b}(y) + \log_{b}(z)
$$

So, the answer is:

$$
\log_{b}(x^{3} y^{5} z) = 3 \log_{b}(x) + 5 \log_{b}(y) + \log_{b}(z)
$$
$$ \log_{b}(x^{3} y^{5} z) = 3 \log_{b}(x) + 5 \log_{b}(y) + \log_{b}(z) $$

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