98. The value of x satisfying the equation `((sqrtpi)^(log_pi(x))).((sqrtpi)^(log_(pi^2)(x))).((sqrtpi).^(log_(pi^4)(x))).((sqrtpi)^(log_(pi^8)(x)))...oo=3` is equal to

Options

A. `sqrt(pi)`
B. `pi`
C. 3
D. `(1)/(3)`

Explanation

To solve the equation

\[ \left(\sqrt{\pi}\right)^{\log_{\pi}(x)} \cdot \left(\sqrt{\pi}\right)^{\log_{\pi^2}(x)} \cdot \left(\sqrt{\pi}\right)^{\log_{\pi^4}(x)} \cdot \left(\sqrt{\pi}\right)^{\log_{\pi^8}(x)} \cdots = 3, \]

we will follow these steps:

### Step 1: Rewrite each term
We can rewrite each term in the product. The first term is:

\[ \left(\sqrt{\pi}\right)^{\log_{\pi}(x)} = \pi^{\frac{1}{2} \log_{\pi}(x)}. \]

Using the property of logarithms, we can express this as:

\[ \pi^{\frac{1}{2} \log_{\pi}(x)} = x^{\frac{1}{2}}. \]

### Step 2: Continue rewriting the terms
Now, consider the second term:

\[ \left(\sqrt{\pi}\right)^{\log_{\pi^2}(x)} = \pi^{\frac{1}{2} \log_{\pi^2}(x)}. \]

Using the change of base formula for logarithms, we have:

\[ \log_{\pi^2}(x) = \frac{1}{2} \log_{\pi}(x), \]

so

\[ \left(\sqrt{\pi}\right)^{\log_{\pi^2}(x)} = \pi^{\frac{1}{2} \cdot \frac{1}{2} \log_{\pi}(x)} = \pi^{\frac{1}{4} \log_{\pi}(x)} = x^{\frac{1}{4}}. \]

### Step 3: Generalize the pattern
Continuing this process, we can see that:

\[ \left(\sqrt{\pi}\right)^{\log_{\pi^4}(x)} = x^{\frac{1}{8}}, \]
\[ \left(\sqrt{\pi}\right)^{\log_{\pi^8}(x)} = x^{\frac{1}{16}}, \]

and so on.

### Step 4: Write the infinite product
Thus, the entire expression can be written as:

\[ x^{\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots}. \]

### Step 5: Calculate the sum of the series
The series \(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots\) is a geometric series with the first term \(a = \frac{1}{2}\) and common ratio \(r = \frac{1}{2}\). The sum of an infinite geometric series is given by:

\[ S = \frac{a}{1 - r} = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = 1. \]

### Step 6: Set up the equation
Now we have:

\[ x^1 = 3. \]

### Step 7: Solve for \(x\)
Thus,

\[ x = 3. \]

### Final Answer
The value of \(x\) satisfying the equation is

\[ \boxed{3}. \]