The pH of a solution is given by the formula:
\[
\text{pH} = -\log [H^+]
\]
We are given that the concentration of hydrogen ions is \( [H^+] = 1 \times 10^{-5} \, \text{mol/L} \).
Now, calculate the pH:
\[
\text{pH} = -\log (1 \times 10^{-5}) = 5
\]
Therefore, the pH of the solution is 5, which corresponds to option (1).