If ` log_(e) log_(5) [sqrt(2x - 2) +3 ] = 0 `

Answer

To solve the equation \( \log_{e} \log_{5} (\sqrt{2x - 2} + 3) = 0 \), we will follow these steps:

### Step 1: Rewrite the logarithmic equation
We start with the equation:
\[ \log_{e} \log_{5} (\sqrt{2x - 2} + 3) = 0 \]
Using the property of logarithms, we know that if \( \log_{a}(b) = 0 \), then \( b = 1 \). Thus, we can rewrite the equation as:
\[ \log_{5} (\sqrt{2x - 2} + 3) = 1 \]

### Step 2: Convert the logarithm to exponential form
Next, we convert the logarithmic equation to its exponential form:
\[ \sqrt{2x - 2} + 3 = 5^{1} \]
This simplifies to:
\[ \sqrt{2x - 2} + 3 = 5 \]

### Step 3: Isolate the square root
Now, we isolate the square root term:
\[ \sqrt{2x - 2} = 5 - 3 \]
This simplifies to:
\[ \sqrt{2x - 2} = 2 \]

### Step 4: Square both sides
Next, we square both sides to eliminate the square root:
\[ 2x - 2 = 2^{2} \]
This simplifies to:
\[ 2x - 2 = 4 \]

### Step 5: Solve for \( x \)
Now, we solve for \( x \):
\[ 2x = 4 + 2 \]
\[ 2x = 6 \]
\[ x = \frac{6}{2} = 3 \]

### Final Answer
Thus, the value of \( x \) that satisfies the equation is:
\[ \boxed{3} \]