To expand the logarithm \(\ln(13x)\), you can use the property of logarithms that states \(\ln(ab) = \ln(a) + \ln(b)\).
Applying this property:
\[ \ln(13x) = \ln(13) + \ln(x) \]
So, the correct response among the options you provided is \(\ln(13) + \ln(x)\), but note that this option does not seem to be listed.
If you need to choose from the provided options, the closest match regarding properties would be terms involving \(\ln(13)\) and \(\ln(x)\).
To clarify the possible responses:
- \( \ln(13 - x) \) does not apply, as that's not equivalent to \(\ln(13x)\).
- \( \frac{\ln 13}{\ln x} \) is incorrect, as it does not represent the logarithm of a product.
- \( \ln 13 - \ln x \) incorrectly suggests subtraction instead of addition.
Thus, none of the provided options accurately represent the correct expansion \(\ln(13) + \ln(x)\).