Solve : `log_(5)(x + 1) - 1 = 1 + log_(5)(x - 1)`.

Answer

To solve the equation \( \log_{5}(x + 1) - 1 = 1 + \log_{5}(x - 1) \), we will follow these steps:

### Step 1: Rewrite the equation
Start by rewriting the equation as it is:
\[ \log_{5}(x + 1) - 1 = 1 + \log_{5}(x - 1) \]

### Step 2: Move logarithmic terms to one side
Rearrange the equation to isolate the logarithmic terms on one side:
\[ \log_{5}(x + 1) - \log_{5}(x - 1) = 1 + 1 \]
This simplifies to:
\[ \log_{5}(x + 1) - \log_{5}(x - 1) = 2 \]

### Step 3: Apply the properties of logarithms
Using the property of logarithms that states \( \log_{b}(a) - \log_{b}(c) = \log_{b}\left(\frac{a}{c}\right) \), we can rewrite the left side:
\[ \log_{5}\left(\frac{x + 1}{x - 1}\right) = 2 \]

### Step 4: Convert to exponential form
Convert the logarithmic equation to its exponential form:
\[ \frac{x + 1}{x - 1} = 5^2 \]
This simplifies to:
\[ \frac{x + 1}{x - 1} = 25 \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiply to solve for \( x \):
\[ x + 1 = 25(x - 1) \]

### Step 6: Distribute and simplify
Distributing on the right side gives:
\[ x + 1 = 25x - 25 \]

### Step 7: Move all terms involving \( x \) to one side
Rearranging the equation to isolate \( x \):
\[ 1 + 25 = 25x - x \]
This simplifies to:
\[ 26 = 24x \]

### Step 8: Solve for \( x \)
Now, divide both sides by 24:
\[ x = \frac{26}{24} \]
Simplifying gives:
\[ x = \frac{13}{12} \]

### Final Answer
Thus, the solution to the equation is:
\[ x = \frac{13}{12} \]