3. Write the following expressions in terms of logs of $$ x, y $$ and $$ z $$. (1) $$ \log x^{2} y $$ (2) $$ \log \frac{r^{3} y^{2}}{z} $$ (3) $$ \log \frac{\sqrt{x} \sqrt[3]{y^{2}}}{z^{4}} $$ (4) $$ \log x y z $$ (5) $$ \log \frac{x}{y z} $$ (6) $$ \log \left(\frac{x}{y}\right)^{2} $$ (7) $$ \log (x y)^{\frac{1}{3}} $$ (8) $$ \log x \sqrt{z} $$ (9) $$ \log \frac{\sqrt[3]{x}}{\sqrt[3]{y z}} $$ (10) $$ \log \sqrt[4]{\frac{x^{3} y^{2}}{z^{4}}} $$ (11) $$ \log x \sqrt{\frac{\sqrt{x}}{z}} $$ (12). $$ \log \sqrt{\frac{x y^{2}}{z^{8}}} $$

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UpStudy AI · Accepted Answer

We will use the properties of logarithms, such as:
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 $$ \log \frac{r^{3} y^{2}}{z} $$:
$$
\log \frac{r^{3} y^{2}}{z} = \log (r^{3} y^{2}) - \log z = \log r^{3} + \log y^{2} - \log z = 3 \log r + 2 \log y - \log z
$$
For $$ \log \frac{\sqrt{x} \sqrt[3]{y^{2}}}{z^{4}} $$:
$$
\log \frac{\sqrt{x} \sqrt[3]{y^{2}}}{z^{4}} = \log (\sqrt{x} \cdot \sqrt[3]{y^{2}}) - \log z^{4} = \log \sqrt{x} + \log \sqrt[3]{y^{2}} - 4 \log z
$$
$$
= \frac{1}{2} \log x + \frac{2}{3} \log y - 4 \log z
$$
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 $$ \log (x y)^{\frac{1}{3}} $$:
$$
\log (x y)^{\frac{1}{3}} = \frac{1}{3} \log (x y) = \frac{1}{3} (\log x + \log y) = \frac{1}{3} \log x + \frac{1}{3} \log y
$$
For $$ \log x \sqrt{z} $$:
$$
\log x \sqrt{z} = \log x + \log \sqrt{z} = \log x + \frac{1}{2} \log z
$$
For $$ \log \frac{\sqrt[3]{x}}{\sqrt[3]{y z}} $$:
$$
\log \frac{\sqrt[3]{x}}{\sqrt[3]{y z}} = \log \sqrt[3]{x} - \log \sqrt[3]{y z} = \frac{1}{3} \log x - \frac{1}{3} (\log y + \log z) = \frac{1}{3} \log x - \frac{1}{3} \log y - \frac{1}{3} \log z
$$
For $$ \log \sqrt[4]{\frac{x^{3} y^{2}}{z^{4}}} $$:
$$
\log \sqrt[4]{\frac{x^{3} y^{2}}{z^{4}}} = \frac{1}{4} \log \frac{x^{3} y^{2}}{z^{4}} = \frac{1}{4} (\log (x^{3} y^{2}) - \log z^{4}) = \frac{1}{4} (3 \log x + 2 \log y - 4 \log z)
$$
$$
= \frac{3}{4} \log x + \frac{1}{2} \log y - \log z
$$
For $$ \log x \sqrt{\frac{\sqrt{x}}{z}} $$:
$$
\log x \sqrt{\frac{\sqrt{x}}{z}} = \log x + \log \sqrt{\frac{\sqrt{x}}{z}} = \log x + \frac{1}{2} \log \frac{\sqrt{x}}{z} = \log x + \frac{1}{2} \left(\frac{1}{2} \log x - \log z\right)
$$
$$
= \log x + \frac{1}{4} \log x - \frac{1}{2} \log z = \frac{5}{4} \log x - \frac{1}{2} \log z
$$
For $$ \log \sqrt{\frac{x y^{2}}{z^{8}}} $$:
$$
\log \sqrt{\frac{x y^{2}}{z^{8}}} = \frac{1}{2} \log \frac{x y^{2}}{z^{8}} = \frac{1}{2} (\log (x y^{2}) - \log z^{8}) = \frac{1}{2} (\log x + 2 \log y - 8 \log z)
$$
$$
= \frac{1}{2} \log x + \log y - 4 \log z
$$
The expressions rewritten in terms of logs of $$ x, y, z $$ are: 1. $$ 2 \log x + \log y $$ 2. $$ 3 \log r + 2 \log y - \log z $$ 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. $$ \frac{1}{3} \log x + \frac{1}{3} \log y $$ 8. $$ \log x + \frac{1}{2} \log z $$ 9. $$ \frac{1}{3} \log x - \frac{1}{3} \log y - \frac{1}{3} \log z $$ 10. $$ \frac{3}{4} \log x + \frac{1}{2} \log y - \log z $$ 11. $$ \frac{5}{4} \log x - \frac{1}{2} \log z $$ 12. $$ \frac{1}{2} \log x + \log y - 4 \log z $$

Write the following expressions in terms of logs of and .
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12).

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Write the following expressions in terms of logs of and . (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12).