if `x >0` , and `log_2 x + log_2 (x^(1/2)) + log_2 (x^(1/4))+ ------------ = 4` then x equals :

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the equation \( \log_2 x + \log_2 (x^{1/2}) + \log_2 (x^{1/4}) + \ldots = 4 \), we will follow these steps:

### Step 1: Rewrite the logarithmic terms
We start by rewriting the logarithmic terms using the property of logarithms \( \log_b (a^m) = m \log_b a \).

\[ \log_2 x + \log_2 (x^{1/2}) + \log_2 (x^{1/4}) + \ldots = \log_2 x + \frac{1}{2} \log_2 x + \frac{1}{4} \log_2 x + \ldots \]

### Step 2: Factor out \( \log_2 x \)
Now, we can factor out \( \log_2 x \) from the left side:

\[ \log_2 x \left( 1 + \frac{1}{2} + \frac{1}{4} + \ldots \right) = 4 \]

### Step 3: Identify the series
The series \( 1 + \frac{1}{2} + \frac{1}{4} + \ldots \) is a geometric series where the first term \( a = 1 \) and the common ratio \( r = \frac{1}{2} \).

### Step 4: Calculate the sum of the series
The sum \( S \) of an infinite geometric series can be calculated using the formula:

\[ S = \frac{a}{1 - r} \]

Substituting the values:

\[ S = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2 \]

### Step 5: Substitute back into the equation
Now we substitute this sum back into our equation:

\[ \log_2 x \cdot 2 = 4 \]

### Step 6: Solve for \( \log_2 x \)
Dividing both sides by 2 gives:

\[ \log_2 x = 2 \]

### Step 7: Convert from logarithmic to exponential form
To find \( x \), we convert from logarithmic form to exponential form:

\[ x = 2^2 = 4 \]

### Final Answer
Thus, the value of \( x \) is:

\[ \boxed{4} \]