3. Write the following expressions in terms of $$ \operatorname{logs} $$ of $$ x, y $$ and $$ z $$. $$ \begin{array}{llll} ext { (1) } \log x^{2} y & ext { (2) } \log \frac{x^{3} y^{2}}{z} & ext { (3) } \log \frac{\sqrt{x} \sqrt[3]{y^{2}}}{z^{4}} & ext { (4) } \log x y z \ ext { (5) } \log \frac{x}{y z} & ext { (6) } \log \left(\frac{x}{y} ight)^{2} & ext { (7) } \log (x y)^{\frac{2}{3}} & ext { (8) } \log x \sqrt{z}\end{array} $$

Question

3. Write the following expressions in terms of $$ \operatorname{logs} $$ of $$ x, y $$ and $$ z $$. $$ \begin{array}{llll}	ext { (1) } \log x^{2} y & 	ext { (2) } \log \frac{x^{3} y^{2}}{z} & 	ext { (3) } \log \frac{\sqrt{x} \sqrt[3]{y^{2}}}{z^{4}} & 	ext { (4) } \log x y z \ 	ext { (5) } \log \frac{x}{y z} & 	ext { (6) } \log \left(\frac{x}{y}ight)^{2} & 	ext { (7) } \log (x y)^{\frac{2}{3}} & 	ext { (8) } \log x \sqrt{z}\end{array} $$

UpStudy AI · Accepted Answer

We will use the properties of logarithms, specifically:
1. $$ \log a^b = b \log a $$
2. $$ \log \frac{a}{b} = \log a - \log b $$
3. $$ \log (a \cdot b) = \log a + \log b $$

Now, let's rewrite each expression accordingly.
For $$ \log x^{2} y $$:
$$
\log x^{2} y = \log x^{2} + \log y = 2 \log x + \log y
$$
For $$ \log \frac{x^{3} y^{2}}{z} $$:
$$
\log \frac{x^{3} y^{2}}{z} = \log (x^{3} y^{2}) - \log z = \log x^{3} + \log y^{2} - \log z = 3 \log x + 2 \log y - \log z
$$
For $$ \log \frac{\sqrt{x} \sqrt[3]{y^{2}}}{z^{4}} $$:
$$
\log \frac{\sqrt{x} \sqrt[3]{y^{2}}}{z^{4}} = \log (\sqrt{x} \sqrt[3]{y^{2}}) - \log z^{4} = \log \sqrt{x} + \log \sqrt[3]{y^{2}} - 4 \log z
$$
$$
= \frac{1}{2} \log x + \frac{2}{3} \log y - 4 \log z
$$
For $$ \log x y z $$:
$$
\log x y z = \log x + \log y + \log z
$$
For $$ \log \frac{x}{y z} $$:
$$
\log \frac{x}{y z} = \log x - \log (y z) = \log x - (\log y + \log z) = \log x - \log y - \log z
$$
For $$ \log \left(\frac{x}{y}\right)^{2} $$:
$$
\log \left(\frac{x}{y}\right)^{2} = 2 \log \frac{x}{y} = 2 (\log x - \log y) = 2 \log x - 2 \log y
$$
For $$ \log (x y)^{\frac{2}{3}} $$:
$$
\log (x y)^{\frac{2}{3}} = \frac{2}{3} \log (x y) = \frac{2}{3} (\log x + \log y) = \frac{2}{3} \log x + \frac{2}{3} \log y
$$
For $$ \log x \sqrt{z} $$:
$$
\log x \sqrt{z} = \log x + \log \sqrt{z} = \log x + \frac{1}{2} \log z
$$
The expressions in terms of logs are: 1. $$ 2 \log x + \log y $$ 2. $$ 3 \log x + 2 \log y - \log z $$ 3. $$ \frac{1}{2} \log x + \frac{2}{3} \log y - 4 \log z $$ 4. $$ \log x + \log y + \log z $$ 5. $$ \log x - \log y - \log z $$ 6. $$ 2 \log x - 2 \log y $$ 7. $$ \frac{2}{3} \log x + \frac{2}{3} \log y $$ 8. $$ \log x + \frac{1}{2} \log z $$

Write the following expressions in terms of of and .

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