Solve for x, if : `log_(x)49 - log_(x)7 + "log"_(x)(1)/(343) + 2 = 0`.

Answer

To solve the equation \( \log_{x}(49) - \log_{x}(7) + \log_{x}\left(\frac{1}{343}\right) + 2 = 0 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \log_{x}(49) - \log_{x}(7) + \log_{x}\left(\frac{1}{343}\right) + 2 = 0 \]
Rearranging gives:
\[ \log_{x}(49) - \log_{x}(7) + \log_{x}\left(\frac{1}{343}\right) = -2 \]

### Step 2: Use logarithmic properties
Using the properties of logarithms, we can combine the logs:
\[ \log_{x}\left(\frac{49 \cdot \frac{1}{343}}{7}\right) = -2 \]
This simplifies to:
\[ \log_{x}\left(\frac{49}{7 \cdot 343}\right) = -2 \]

### Step 3: Simplify the expression inside the logarithm
Calculating \( \frac{49}{7 \cdot 343} \):
\[ 7 \cdot 343 = 7 \cdot 7^3 = 7^4 = 2401 \]
Thus, we have:
\[ \frac{49}{2401} = \frac{7^2}{7^4} = \frac{1}{7^2} = \frac{1}{49} \]
So we can rewrite the equation as:
\[ \log_{x}\left(\frac{1}{49}\right) = -2 \]

### Step 4: Rewrite the logarithm
We know that:
\[ \log_{x}\left(\frac{1}{49}\right) = \log_{x}(49^{-1}) = -\log_{x}(49) \]
Thus, we have:
\[ -\log_{x}(49) = -2 \]
This simplifies to:
\[ \log_{x}(49) = 2 \]

### Step 5: Convert the logarithmic equation to exponential form
From \( \log_{x}(49) = 2 \), we can convert this to exponential form:
\[ x^2 = 49 \]

### Step 6: Solve for \( x \)
Taking the square root of both sides gives:
\[ x = 7 \quad \text{or} \quad x = -7 \]
Since the base of a logarithm must be positive, we take:
\[ x = 7 \]

### Final Answer
Thus, the solution for \( x \) is:
\[ \boxed{7} \]