Sum of all the values of x satisfying the equation
`log_(17) log_(11) (sqrt(x + 11) + sqrt(x)) = 0" "` (1) is

Options

A. 25
B. 36
C. 171
D. 0

Answer

To solve the equation \( \log_{17} \log_{11} (\sqrt{x + 11} + \sqrt{x}) = 0 \), we will follow these steps:

### Step 1: Simplify the logarithmic equation
Using the property of logarithms, we know that if \( \log_a b = 0 \), then \( b = 1 \). Therefore, we can rewrite the equation as:
\[ \log_{11} (\sqrt{x + 11} + \sqrt{x}) = 1 \]

### Step 2: Exponentiate to eliminate the logarithm
Next, we exponentiate both sides to remove the logarithm:
\[ \sqrt{x + 11} + \sqrt{x} = 11 \]

### Step 3: Isolate one of the square roots
Now, we can isolate one of the square roots:
\[ \sqrt{x + 11} = 11 - \sqrt{x} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives us:
\[ x + 11 = (11 - \sqrt{x})^2 \]

### Step 5: Expand the right-hand side
Expanding the right-hand side:
\[ x + 11 = 121 - 22\sqrt{x} + x \]

### Step 6: Simplify the equation
Now, we can simplify the equation:
\[ 11 = 121 - 22\sqrt{x} \]
\[ 22\sqrt{x} = 121 - 11 \]
\[ 22\sqrt{x} = 110 \]

### Step 7: Solve for \(\sqrt{x}\)
Dividing both sides by 22:
\[ \sqrt{x} = \frac{110}{22} = 5 \]

### Step 8: Square to find \(x\)
Now, squaring both sides gives us:
\[ x = 5^2 = 25 \]

### Conclusion
The only value of \(x\) that satisfies the original equation is \(x = 25\). Therefore, the sum of all values of \(x\) satisfying the equation is:
\[ \text{Sum} = 25 \]