Use properties of logarithms to evaluate without using a calculator. $$ 8 \log _{4}(2)+\frac{\log _{4}(64)}{3 \log _{4}(4)} $$

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UpStudy AI · Accepted Answer

To evaluate the expression without a calculator:

$$
8 \log _{4}(2) + \frac{\log _{4}(64)}{3 \log _{4}(4)}
$$

Let's break it down step by step.

1. **Evaluate $$\log_4(2)$$:**

Recall that $$4 = 2^2$$. Using the change of base formula:
   
   $$
   \log_4(2) = \frac{\log_2(2)}{\log_2(4)} = \frac{1}{2}
   $$
   
2. **Evaluate $$\log_4(64)$$:**

Since $$64 = 2^6$$ and $$4 = 2^2$$:
   
   $$
   \log_4(64) = \frac{\log_2(64)}{\log_2(4)} = \frac{6}{2} = 3
   $$
   
3. **Evaluate $$\log_4(4)$$:**

$$
   \log_4(4) = 1
   $$
   
4. **Substitute these values back into the original expression:**

$$
   8 \cdot \frac{1}{2} + \frac{3}{3 \cdot 1} = 4 + 1 = 5
   $$

**Final Answer:**  
5
The expression evaluates to 5.

Use properties of logarithms to evaluate without using a calculator.

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