Whenever a,b, and c are positive real numbers, which of the following expressions is equivalent to `log_4 u-2 log_8 b+1/2 log_4 c` ?

Options

A. `log_4 asqrtc-log_8 b^2`
B. `log_4 (ac)/2-log_8 2b`
C. `log_4 (asqrtc)/b`
D. `log_4 (a-c)-log_8 2b`

Answer

To simplify the expression \( \log_4 a - 2 \log_8 b + \frac{1}{2} \log_4 c \), we will follow these steps:

### Step 1: Rewrite the logarithmic terms
The expression is:
\[ \log_4 a - 2 \log_8 b + \frac{1}{2} \log_4 c \]

### Step 2: Convert \( 2 \log_8 b \) to base 4
Using the change of base formula, we can convert \( \log_8 b \) to base 4:
\[ \log_8 b = \frac{\log_4 b}{\log_4 8} \]
Since \( \log_4 8 = \log_4 (2^3) = 3 \log_4 2 \), we have:
\[ \log_8 b = \frac{\log_4 b}{3 \log_4 2} \]
Thus,
\[ 2 \log_8 b = 2 \cdot \frac{\log_4 b}{3 \log_4 2} = \frac{2 \log_4 b}{3 \log_4 2} \]

### Step 3: Rewrite \( \frac{1}{2} \log_4 c \)
The term \( \frac{1}{2} \log_4 c \) can be rewritten as:
\[ \frac{1}{2} \log_4 c = \log_4 (c^{1/2}) = \log_4 (\sqrt{c}) \]

### Step 4: Combine the logarithmic terms
Now we can combine the terms:
\[ \log_4 a + \log_4 (\sqrt{c}) - \frac{2 \log_4 b}{3 \log_4 2} \]
The first two terms can be combined using the property of logarithms:
\[ \log_4 (a \cdot \sqrt{c}) = \log_4 (a \sqrt{c}) \]

### Step 5: Rewrite the expression
Now we have:
\[ \log_4 (a \sqrt{c}) - \frac{2 \log_4 b}{3 \log_4 2} \]

### Step 6: Simplify the logarithm with base 8
We can express \( \log_4 b^2 \) in terms of base 8:
\[ \log_4 b^2 = 2 \log_4 b = \frac{2 \log_4 b}{3 \log_4 2} \cdot 3 \log_4 2 \]
Thus, we can rewrite the entire expression as:
\[ \log_4 (a \sqrt{c}) - \log_8 (b^2) \]

### Final Expression
The final expression is:
\[ \log_4 (a \sqrt{c}) - \log_8 (b^2) \]

### Conclusion
We can conclude that the simplified expression is:
\[ \log_4 (a \sqrt{c}) - \log_8 (b^2) \]