Voluntary worksheet focusing on expanding logarithmic expressions using logarithmic properties.

Let's solve each logarithmic expression by expanding them using the properties of logarithms.

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🔷 Logarithmic Properties Used:

1. Product Rule:
$\log_b(MN) = \log_b M + \log_b N$

2. Quotient Rule:
$\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$

3. Power Rule:
$\log_b(M^n) = n \log_b M$

4. Roots as Powers:
$\sqrt[n]{M} = M^{1/n}$, so $\log_b(\sqrt[n]{M}) = \frac{1}{n} \log_b M$

5. Log of a power inside:
$\log_b(M^k) = k \log_b M$, even if $M$ is a product or quotient.

6. Change of base (if needed): Not required here.

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Now, let’s expand each one step-by-step.

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1) $\ln(x^6 y^3)$

Use product rule:
$$
\ln(x^6 y^3) = \ln(x^6) + \ln(y^3)
$$
Apply power rule:
$$
= 6\ln x + 3\ln y
$$

✔ Answer: $6\ln x + 3\ln y$

---

2) $\log_8(x \cdot y \cdot z^3)$

Use product rule:
$$
= \log_8 x + \log_8 y + \log_8(z^3)
$$
Apply power rule to last term:
$$
= \log_8 x + \log_8 y + 3\log_8 z
$$

✔ Answer: $\log_8 x + \log_8 y + 3\log_8 z$

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3) $\log_9\left(\frac{3^3}{7}\right)^4$

First, apply power rule to the entire expression:
$$
= 4 \log_9\left(\frac{3^3}{7}\right)
$$
Now use quotient rule:
$$
= 4 \left( \log_9(3^3) - \log_9(7) \right)
$$
Apply power rule to $\log_9(3^3)$:
$$
= 4 \left( 3\log_9 3 - \log_9 7 \right)
$$
Note: $\log_9 3 = \log_9(9^{1/2}) = \frac{1}{2}$, since $9^{1/2} = 3$

So:
$$
= 4 \left( 3 \cdot \frac{1}{2} - \log_9 7 \right) = 4 \left( \frac{3}{2} - \log_9 7 \right)
$$
$$
= 6 - 4\log_9 7
$$

✔ Answer: $6 - 4\log_9 7$

---

4) $\log_7\left(\frac{x^3}{y}\right)^3$

Apply power rule:
$$
= 3 \log_7\left(\frac{x^3}{y}\right)
$$
Apply quotient rule:
$$
= 3 \left( \log_7(x^3) - \log_7 y \right)
$$
Apply power rule:
$$
= 3 \left( 3\log_7 x - \log_7 y \right) = 9\log_7 x - 3\log_7 y
$$

✔ Answer: $9\log_7 x - 3\log_7 y$

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5) $\log_8(a^6 b^5)$

Use product rule:
$$
= \log_8(a^6) + \log_8(b^5)
$$
Apply power rule:
$$
= 6\log_8 a + 5\log_8 b
$$

✔ Answer: $6\log_8 a + 5\log_8 b$

---

6) $\log_4(6^3 \cdot 11^3)$

Use product rule:
$$
= \log_4(6^3) + \log_4(11^3)
$$
Apply power rule:
$$
= 3\log_4 6 + 3\log_4 11
$$
Factor out 3:
$$
= 3(\log_4 6 + \log_4 11)
$$

✔ Answer: $3(\log_4 6 + \log_4 11)$

---

7) $\log_3\left(\frac{u^3}{v}\right)^2$

Apply power rule:
$$
= 2 \log_3\left(\frac{u^3}{v}\right)
$$
Apply quotient rule:
$$
= 2 \left( \log_3(u^3) - \log_3 v \right)
$$
Apply power rule:
$$
= 2(3\log_3 u - \log_3 v) = 6\log_3 u - 2\log_3 v
$$

✔ Answer: $6\log_3 u - 2\log_3 v$

---

8) $\ln \sqrt[3]{u \cdot v \cdot w}$

Rewrite root as exponent:
$$
= \ln \left( (u v w)^{1/3} \right)
$$
Apply power rule:
$$
= \frac{1}{3} \ln(u v w)
$$
Use product rule:
$$
= \frac{1}{3} (\ln u + \ln v + \ln w)
$$

✔ Answer: $\frac{1}{3}(\ln u + \ln v + \ln w)$

---

9) $\log_6(3 \cdot 2 \cdot 5^6)$

Use product rule:
$$
= \log_6 3 + \log_6 2 + \log_6(5^6)
$$
Apply power rule:
$$
= \log_6 3 + \log_6 2 + 6\log_6 5
$$

✔ Answer: $\log_6 3 + \log_6 2 + 6\log_6 5$

---

10) $\log_4(2 \cdot 11 \cdot 7^4)$

Use product rule:
$$
= \log_4 2 + \log_4 11 + \log_4(7^4)
$$
Apply power rule:
$$
= \log_4 2 + \log_4 11 + 4\log_4 7
$$

Note: $\log_4 2 = \log_4(4^{1/2}) = \frac{1}{2}$

So:
$$
= \frac{1}{2} + \log_4 11 + 4\log_4 7
$$

✔ Answer: $\frac{1}{2} + \log_4 11 + 4\log_4 7$

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11) $\log_6(c^5 \sqrt[3]{a})$

Rewrite root:
$$
= \log_6(c^5 \cdot a^{1/3})
$$
Use product rule:
$$
= \log_6(c^5) + \log_6(a^{1/3})
$$
Apply power rule:
$$
= 5\log_6 c + \frac{1}{3} \log_6 a
$$

✔ Answer: $5\log_6 c + \frac{1}{3} \log_6 a$

---

12) $\ln\left(\frac{5^2}{2}\right)^5$

Apply power rule:
$$
= 5 \ln\left(\frac{25}{2}\right)
$$
Apply quotient rule:
$$
= 5 (\ln 25 - \ln 2)
$$
But $\ln 25 = \ln(5^2) = 2\ln 5$, so:
$$
= 5 (2\ln 5 - \ln 2) = 10\ln 5 - 5\ln 2
$$

✔ Answer: $10\ln 5 - 5\ln 2$

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13) $\log_5\left(\frac{x^3}{y}\right)^6$

Apply power rule:
$$
= 6 \log_5\left(\frac{x^3}{y}\right)
$$
Apply quotient rule:
$$
= 6 (\log_5(x^3) - \log_5 y)
$$
Apply power rule:
$$
= 6(3\log_5 x - \log_5 y) = 18\log_5 x - 6\log_5 y
$$

✔ Answer: $18\log_5 x - 6\log_5 y$

---

14) $\log_4(7^3 \sqrt[3]{2})$

Rewrite cube root:
$$
= \log_4(7^3 \cdot 2^{1/3})
$$
Use product rule:
$$
= \log_4(7^3) + \log_4(2^{1/3})
$$
Apply power rule:
$$
= 3\log_4 7 + \frac{1}{3} \log_4 2
$$
Note: $\log_4 2 = \frac{1}{2}$, so:
$$
= 3\log_4 7 + \frac{1}{3} \cdot \frac{1}{2} = 3\log_4 7 + \frac{1}{6}
$$

✔ Answer: $3\log_4 7 + \frac{1}{6}$

---

15) $\log_2(u \cdot v \cdot w^2)$

Use product rule:
$$
= \log_2 u + \log_2 v + \log_2(w^2)
$$
Apply power rule:
$$
= \log_2 u + \log_2 v + 2\log_2 w
$$

✔ Answer: $\log_2 u + \log_2 v + 2\log_2 w$

---

16) $\log_9(12^3 \cdot 7)^6$

First, simplify inside:
$$
= \log_9\left((12^3 \cdot 7)^6\right)
$$
Apply power rule:
$$
= 6 \log_9(12^3 \cdot 7)
$$
Use product rule:
$$
= 6 \left( \log_9(12^3) + \log_9 7 \right)
$$
Apply power rule:
$$
= 6 (3\log_9 12 + \log_9 7) = 18\log_9 12 + 6\log_9 7
$$

✔ Answer: $18\log_9 12 + 6\log_9 7$

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17) $\log_9(c^5 \sqrt[3]{a})$

Same as #11:
$$
= \log_9(c^5 \cdot a^{1/3}) = \log_9(c^5) + \log_9(a^{1/3})
$$
$$
= 5\log_9 c + \frac{1}{3} \log_9 a
$$

✔ Answer: $5\log_9 c + \frac{1}{3} \log_9 a$

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✔ Final Answers Summary:

| Problem | Expanded Form |
|--------|----------------|
| 1 | $6\ln x + 3\ln y$ |
| 2 | $\log_8 x + \log_8 y + 3\log_8 z$ |
| 3 | $6 - 4\log_9 7$ |
| 4 | $9\log_7 x - 3\log_7 y$ |
| 5 | $6\log_8 a + 5\log_8 b$ |
| 6 | $3(\log_4 6 + \log_4 11)$ |
| 7 | $6\log_3 u - 2\log_3 v$ |
| 8 | $\frac{1}{3}(\ln u + \ln v + \ln w)$ |
| 9 | $\log_6 3 + \log_6 2 + 6\log_6 5$ |
| 10 | $\frac{1}{2} + \log_4 11 + 4\log_4 7$ |
| 11 | $5\log_6 c + \frac{1}{3} \log_6 a$ |
| 12 | $10\ln 5 - 5\ln 2$ |
| 13 | $18\log_5 x - 6\log_5 y$ |
| 14 | $3\log_4 7 + \frac{1}{6}$ |
| 15 | $\log_2 u + \log_2 v + 2\log_2 w$ |
| 16 | $18\log_9 12 + 6\log_9 7$ |
| 17 | $5\log_9 c + \frac{1}{3} \log_9 a$ |

Let me know if you'd like these condensed back into single logs!