Expand each logarithm. 1) $$ \log (6 \cdot 11) $$ 2) $$ \log (5 \cdot 3) $$ 3) $$ \log \left(\frac{6}{11}\right)^{5} $$ 4) $$ \log \left(3 \cdot 2^{3}\right) $$ 5) $$ \log \frac{2^{4}}{5} $$ 6) $$ \log \left(\frac{6}{5}\right)^{6} $$ 7) $$ \log \frac{x}{y^{6}} $$ 8) $$ \log (a \cdot b)^{2} $$ 9) $$ \log \frac{u^{4}}{v} $$ 10) $$ \log \frac{x}{y^{5}} $$ 11) $$ \log \sqrt[3]{x \cdot y \cdot z} $$ 12) $$ \log \left(x \cdot y \cdot z^{2}\right) $$

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UpStudy AI · Accepted Answer

Certainly! Let's expand each logarithmic expression using logarithm properties:

### 1) $$ \log (6 \cdot 11) $$
**Property Used:** $$ \log(ab) = \log a + \log b $$

**Expansion:**
$$
\log 6 + \log 11
$$

---

### 2) $$ \log (5 \cdot 3) $$
**Property Used:** $$ \log(ab) = \log a + \log b $$

**Expansion:**
$$
\log 5 + \log 3
$$

---

### 3) $$ \log \left(\frac{6}{11}\right)^{5} $$
**Properties Used:**
- $$ \log(a^k) = k \log a $$
- $$ \log\left(\frac{a}{b}\right) = \log a - \log b $$

**Expansion:**
$$
5 \left( \log 6 - \log 11 \right) = 5 \log 6 - 5 \log 11
$$

---

### 4) $$ \log \left(3 \cdot 2^{3}\right) $$
**Properties Used:**
- $$ \log(ab) = \log a + \log b $$
- $$ \log(a^k) = k \log a $$

**Expansion:**
$$
\log 3 + 3 \log 2
$$

---

### 5) $$ \log \frac{2^{4}}{5} $$
**Properties Used:**
- $$ \log\left(\frac{a}{b}\right) = \log a - \log b $$
- $$ \log(a^k) = k \log a $$

**Expansion:**
$$
4 \log 2 - \log 5
$$

---

### 6) $$ \log \left(\frac{6}{5}\right)^{6} $$
**Properties Used:**
- $$ \log(a^k) = k \log a $$
- $$ \log\left(\frac{a}{b}\right) = \log a - \log b $$

**Expansion:**
$$
6 \left( \log 6 - \log 5 \right) = 6 \log 6 - 6 \log 5
$$

---

### 7) $$ \log \frac{x}{y^{6}} $$
**Properties Used:**
- $$ \log\left(\frac{a}{b}\right) = \log a - \log b $$
- $$ \log(b^k) = k \log b $$

**Expansion:**
$$
\log x - 6 \log y
$$

---

### 8) $$ \log (a \cdot b)^{2} $$
**Properties Used:**
- $$ \log(a^k) = k \log a $$
- $$ \log(ab) = \log a + \log b $$

**Expansion:**
$$
2 \log (a \cdot b) = 2 \log a + 2 \log b
$$

---

### 9) $$ \log \frac{u^{4}}{v} $$
**Properties Used:**
- $$ \log\left(\frac{a}{b}\right) = \log a - \log b $$
- $$ \log(a^k) = k \log a $$

**Expansion:**
$$
4 \log u - \log v
$$

---

### 10) $$ \log \frac{x}{y^{5}} $$
**Properties Used:**
- $$ \log\left(\frac{a}{b}\right) = \log a - \log b $$
- $$ \log(b^k) = k \log b $$

**Expansion:**
$$
\log x - 5 \log y
$$

---

### 11) $$ \log \sqrt[3]{x \cdot y \cdot z} $$
**Properties Used:**
- $$ \sqrt[3]{a} = a^{1/3} $$
- $$ \log(a^k) = k \log a $$
- $$ \log(abc) = \log a + \log b + \log c $$

**Expansion:**
$$
\log (x \cdot y \cdot z)^{1/3} = \frac{1}{3} \log x + \frac{1}{3} \log y + \frac{1}{3} \log z
$$

---

### 12) $$ \log \left(x \cdot y \cdot z^{2}\right) $$
**Properties Used:**
- $$ \log(ab) = \log a + \log b + \log c $$ (for three terms)
- $$ \log(a^k) = k \log a $$

**Expansion:**
$$
\log x + \log y + 2 \log z
$$

---

Each expansion utilizes fundamental logarithmic identities to break down the expressions into simpler logarithmic terms.
Here are the expanded logarithmic expressions: 1. $$ \log 6 + \log 11 $$ 2. $$ \log 5 + \log 3 $$ 3. $$ 5 \log 6 - 5 \log 11 $$ 4. $$ \log 3 + 3 \log 2 $$ 5. $$ 4 \log 2 - \log 5 $$ 6. $$ 6 \log 6 - 6 \log 5 $$ 7. $$ \log x - 6 \log y $$ 8. $$ 2 \log a + 2 \log b $$ 9. $$ 4 \log u - \log v $$ 10. $$ \log x - 5 \log y $$ 11. $$ \frac{1}{3} \log x + \frac{1}{3} \log y + \frac{1}{3} \log z $$ 12. $$ \log x + \log y + 2 \log z $$

Expand each logarithm.

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