The solution of `log_(7)log_(5)(sqrt((x+5))+sqrt(x)]=0` is …...

Answer

To solve the equation \( \log_{7}(\log_{5}(\sqrt{x+5} + \sqrt{x})) = 0 \), we will follow these steps:

### Step 1: Simplify the logarithmic equation
We start with the equation:
\[ \log_{7}(\log_{5}(\sqrt{x+5} + \sqrt{x})) = 0 \]
Using the property of logarithms, \( \log_{a}(b) = 0 \) implies that \( b = 1 \). Therefore, we can rewrite our equation as:
\[ \log_{5}(\sqrt{x+5} + \sqrt{x}) = 1 \]

### Step 2: Exponentiate to remove the logarithm
Next, we exponentiate both sides using base 5:
\[ \sqrt{x+5} + \sqrt{x} = 5^{1} = 5 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation:
\[ \sqrt{x+5} = 5 - \sqrt{x} \]

### Step 4: Square both sides
To eliminate the square root, we square both sides:
\[ x + 5 = (5 - \sqrt{x})^2 \]
Expanding the right-hand side:
\[ x + 5 = 25 - 10\sqrt{x} + x \]

### Step 5: Simplify the equation
Now, we can simplify this equation:
\[ x + 5 = 25 - 10\sqrt{x} + x \]
Subtract \( x \) from both sides:
\[ 5 = 25 - 10\sqrt{x} \]
Now, subtract 25 from both sides:
\[ -20 = -10\sqrt{x} \]

### Step 6: Solve for \(\sqrt{x}\)
Dividing both sides by -10 gives:
\[ \sqrt{x} = 2 \]

### Step 7: Square both sides again
Now, we square both sides to find \( x \):
\[ x = 2^2 = 4 \]

### Conclusion
Thus, the solution to the equation is:
\[ \boxed{4} \]