Expand each as much as possible. \#6: $$ \log _{3}(3 x) $$ $$ \# 7: \log _{2}(x(x+1)) $$ $$ \# 8: \log \left(x^{\wedge} 5^{*} y^{\wedge} 6 / z^{\wedge} 4\right) $$

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Sure, let's expand each logarithmic expression step by step using logarithm properties. The primary properties we'll use are:

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:** $$\log_b (M^k) = k \log_b M$$

---

### **Problem #6: $$ \log_{3}(3x) $$**

**Step 1: Apply the Product Rule**

$$
\log_{3}(3x) = \log_{3}3 + \log_{3}x
$$

**Step 2: Simplify $$\log_{3}3$$**

Since $$3^1 = 3$$, we have:

$$
\log_{3}3 = 1
$$

**Final Expanded Form:**

$$
\log_{3}(3x) = 1 + \log_{3}x
$$

---

### **Problem #7: $$ \log_{2}(x(x+1)) $$**

**Step 1: Apply the Product Rule**

$$
\log_{2}(x(x+1)) = \log_{2}x + \log_{2}(x+1)
$$

**Final Expanded Form:**

$$
\log_{2}(x(x+1)) = \log_{2}x + \log_{2}(x+1)
$$

*Note:* This is as far as we can expand the expression without additional information about the relationship between $$x$$ and $$x+1$$.

---

### **Problem #8: $$ \log\left(\frac{x^{5} \cdot y^{6}}{z^{4}}\right) $$**

**Step 1: Apply the Quotient Rule**

$$
\log\left(\frac{x^{5} \cdot y^{6}}{z^{4}}\right) = \log(x^{5} \cdot y^{6}) - \log(z^{4})
$$

**Step 2: Apply the Product Rule to $$\log(x^{5} \cdot y^{6})$$**

$$
\log(x^{5} \cdot y^{6}) = \log x^{5} + \log y^{6}
$$

**Step 3: Apply the Power Rule to Each Term**

$$
\log x^{5} = 5 \log x
$$
$$
\log y^{6} = 6 \log y
$$
$$
\log z^{4} = 4 \log z
$$

**Final Expanded Form:**

$$
\log\left(\frac{x^{5} \cdot y^{6}}{z^{4}}\right) = 5 \log x + 6 \log y - 4 \log z
$$

---

**Summary of Expanded Forms:**

1. **#6:**  
   $$
   \log_{3}(3x) = 1 + \log_{3}x
   $$

2. **#7:**  
   $$
   \log_{2}(x(x+1)) = \log_{2}x + \log_{2}(x+1)
   $$

3. **#8:**  
   $$
   \log\left(\frac{x^{5} \cdot y^{6}}{z^{4}}\right) = 5 \log x + 6 \log y - 4 \log z
   $$

---

These expansions break down the original logarithmic expressions into simpler components, making them easier to work with in various mathematical contexts.
$$ \log_{3}(3x) = 1 + \log_{3}x $$ $$ \log_{2}(x(x+1)) = \log_{2}x + \log_{2}(x+1) $$ $$ \log\left(\frac{x^{5} \cdot y^{6}}{z^{4}}\right) = 5 \log x + 6 \log y - 4 \log z $$

Expand each as much as possible.
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