If `(1)/(log_(3)pi)+(1)/(log_(4)pi)gtx`, then `x` be

Options

A. `2`
B. `3`
C. `3*5`
D. `pi`

Answer

To solve the inequality \( \frac{1}{\log_{3}\pi} + \frac{1}{\log_{4}\pi} > x \), we will follow these steps:

### Step 1: Use the Change of Base Formula
We start by applying the change of base formula for logarithms, which states that:

\[ \log_{a}b = \frac{1}{\log_{b}a} \]

Using this property, we can rewrite our expression:

\[ \frac{1}{\log_{3}\pi} = \log_{\pi}3 \quad \text{and} \quad \frac{1}{\log_{4}\pi} = \log_{\pi}4 \]

Thus, we have:

\[ \log_{\pi}3 + \log_{\pi}4 > x \]

### Step 2: Combine the Logarithms
Using the property of logarithms that states \( \log_{a}b + \log_{a}c = \log_{a}(bc) \), we can combine the logarithms:

\[ \log_{\pi}(3 \cdot 4) > x \]

This simplifies to:

\[ \log_{\pi}12 > x \]

### Step 3: Take the Antilogarithm
To find the value of \( x \), we can rewrite the inequality in exponential form:

\[ 12 > \pi^x \]

### Step 4: Solve for \( x \)
Now, we need to find \( x \) such that \( \pi^x < 12 \). We can estimate the value of \( \pi \):

\[ \pi \approx 3.14 \]

Calculating powers of \( \pi \):

- \( \pi^2 \approx 3.14^2 \approx 9.86 \)
- \( \pi^3 \approx 3.14^3 \approx 31.006 \)

From this, we can see that \( \pi^2 < 12 < \pi^3 \). Therefore, \( x \) must be between 2 and 3.

### Step 5: Determine the Maximum Integer Value of \( x \)
Since \( x \) must be less than 3 and greater than 2, the largest integer value \( x \) can take is:

\[ x = 2 \]

### Conclusion
Thus, the value of \( x \) that satisfies the inequality is:

\[ \boxed{2} \]