`y = log_sqrt3 81 sqrt3 + log_(1/4)32`

Options

A. `11/2`
B. `13/2`
C. `15/2`
D. `17/2`

Answer

To solve the expression \( y = \log_{\sqrt{3}} (81 \sqrt{3}) + \log_{\frac{1}{4}} (32) \), we will simplify each logarithmic term step by step.

### Step 1: Simplify the first logarithm
We start with the first term:
\[ \log_{\sqrt{3}} (81 \sqrt{3}) \]
Using the property of logarithms that states \( \log_a (bc) = \log_a b + \log_a c \), we can separate this into two parts:
\[ \log_{\sqrt{3}} (81) + \log_{\sqrt{3}} (\sqrt{3}) \]

### Step 2: Simplify \( \log_{\sqrt{3}} (\sqrt{3}) \)
We know that:
\[ \log_{\sqrt{3}} (\sqrt{3}) = 1 \]
because any logarithm of a number to its own base is 1.

### Step 3: Simplify \( \log_{\sqrt{3}} (81) \)
Next, we simplify \( \log_{\sqrt{3}} (81) \). We can express 81 as a power of 3:
\[ 81 = 3^4 \]
Thus,
\[ \log_{\sqrt{3}} (81) = \log_{\sqrt{3}} (3^4) \]
Using the power property of logarithms \( \log_a (b^c) = c \cdot \log_a (b) \):
\[ \log_{\sqrt{3}} (3^4) = 4 \cdot \log_{\sqrt{3}} (3) \]

### Step 4: Simplify \( \log_{\sqrt{3}} (3) \)
Now, we simplify \( \log_{\sqrt{3}} (3) \). We can express this as:
\[ \log_{\sqrt{3}} (3) = \frac{\log_{3} (3)}{\log_{3} (\sqrt{3})} \]
Since \( \log_{3} (3) = 1 \) and \( \log_{3} (\sqrt{3}) = \frac{1}{2} \), we have:
\[ \log_{\sqrt{3}} (3) = \frac{1}{\frac{1}{2}} = 2 \]
Thus:
\[ \log_{\sqrt{3}} (81) = 4 \cdot 2 = 8 \]

### Step 5: Combine the first logarithm
Now we can combine our results:
\[ \log_{\sqrt{3}} (81 \sqrt{3}) = 8 + 1 = 9 \]

### Step 6: Simplify the second logarithm
Now we move to the second term:
\[ \log_{\frac{1}{4}} (32) \]
We can express \( \frac{1}{4} \) as \( 2^{-2} \) and \( 32 \) as \( 2^5 \):
\[ \log_{2^{-2}} (2^5) \]
Using the change of base formula:
\[ \log_{2^{-2}} (2^5) = \frac{5}{-2} = -\frac{5}{2} \]

### Step 7: Combine both logarithms
Now we combine both parts:
\[ y = 9 - \frac{5}{2} \]
To combine these, we convert 9 into a fraction:
\[ 9 = \frac{18}{2} \]
Thus:
\[ y = \frac{18}{2} - \frac{5}{2} = \frac{13}{2} \]

### Final Answer
The final answer is:
\[ y = \frac{13}{2} \]