White solving logarithmic equations or logarithmic
inequalities care must be taken to ensure that the value
of the variable obtained do indeed satisfy the given
equation . Often the solution consists in transforming the original equation to form which can be solved with ease .
But in bargain the process the transformations carried
out are hot always equivalent .In what follows one must
verify that the values of veriables obtained indeed satisfy
original equation or inequation.
Let S be the set of all solutions x in real numbers of
the equation ` (log_(5) x)^(2) + log_(5x) (5)/(x) = 1` . Then ` sum _( x in S) x `

Options

A. 126
B. `(151)/(25)`
C. `(131)/(25)`
D. 131

Answer

To solve the equation \( (\log_{5} x)^{2} + \frac{\log_{5} (5)}{x} = 1 \), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Rewrite the Equation**:
Start with the given equation:
\[ (\log_{5} x)^{2} + \frac{\log_{5} (5)}{x} = 1 \]
Since \(\log_{5} (5) = 1\), we can simplify the equation to:
\[ (\log_{5} x)^{2} + \frac{1}{x} = 1 \]

2. **Substitute**:
Let \( t = \log_{5} x \). Then, we can express \( x \) in terms of \( t \):
\[ x = 5^{t} \]
Substitute this into the equation:
\[ t^{2} + \frac{1}{5^{t}} = 1 \]

3. **Rearrange the Equation**:
Rearranging gives:
\[ t^{2} = 1 - \frac{1}{5^{t}} \]
Multiply through by \( 5^{t} \) to eliminate the fraction:
\[ t^{2} \cdot 5^{t} = 5^{t} - 1 \]
Rearranging this gives:
\[ t^{2} \cdot 5^{t} + 1 - 5^{t} = 0 \]
or
\[ t^{2} \cdot 5^{t} - 5^{t} + 1 = 0 \]

4. **Factor the Equation**:
Factor out \( 5^{t} \):
\[ 5^{t}(t^{2} - 1) + 1 = 0 \]
This implies:
\[ t^{2} - 1 = 0 \]

5. **Solve for \( t \)**:
The equation \( t^{2} - 1 = 0 \) can be factored as:
\[ (t - 1)(t + 1) = 0 \]
Thus, \( t = 1 \) or \( t = -1 \).

6. **Convert Back to \( x \)**:
Recall that \( t = \log_{5} x \):
- For \( t = 1 \):
\[ \log_{5} x = 1 \implies x = 5^{1} = 5 \]
- For \( t = -1 \):
\[ \log_{5} x = -1 \implies x = 5^{-1} = \frac{1}{5} \]

7. **Set of Solutions**:
The solutions are \( x = 5 \) and \( x = \frac{1}{5} \).

8. **Sum of Solutions**:
Now, we need to find the sum of all solutions in the set \( S \):
\[ \text{Sum} = 5 + \frac{1}{5} = 5 + 0.2 = 5.2 \]

### Final Answer:
The sum of all solutions \( x \) in the set \( S \) is:
\[ \frac{26}{5} \]