If `"log"_(2)x + "log"_(4)x + "log"_(16)x = (21)/(4)`, then x equals to

Options

A. 8
B. 4
C. 2
D. 16

Answer

To solve the equation \( \log_2 x + \log_4 x + \log_{16} x = \frac{21}{4} \), we can follow these steps:

### Step 1: Rewrite the logarithms in terms of base 2
We know that:
- \( \log_4 x = \log_{2^2} x = \frac{1}{2} \log_2 x \)
- \( \log_{16} x = \log_{2^4} x = \frac{1}{4} \log_2 x \)

Thus, we can rewrite the equation as:
\[ \log_2 x + \frac{1}{2} \log_2 x + \frac{1}{4} \log_2 x = \frac{21}{4} \]

### Step 2: Combine the logarithmic terms
Let \( t = \log_2 x \). Then, we can express the equation as:
\[ t + \frac{1}{2} t + \frac{1}{4} t = \frac{21}{4} \]

### Step 3: Find a common denominator
The common denominator for the left side is 4. Thus, we can rewrite the equation:
\[ \frac{4t}{4} + \frac{2t}{4} + \frac{t}{4} = \frac{21}{4} \]
This simplifies to:
\[ \frac{4t + 2t + t}{4} = \frac{21}{4} \]
\[ \frac{7t}{4} = \frac{21}{4} \]

### Step 4: Eliminate the denominator
Multiplying both sides by 4 gives:
\[ 7t = 21 \]

### Step 5: Solve for \( t \)
Dividing both sides by 7 yields:
\[ t = 3 \]

### Step 6: Substitute back to find \( x \)
Since \( t = \log_2 x \), we have:
\[ \log_2 x = 3 \]
This implies:
\[ x = 2^3 = 8 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{8} \]