To solve this problem, we need to evaluate the given logarithmic expressions and determine which of the provided inequalities is correct. Let's analyze each expression:

We are given:

• \(a = 30!\), \(b = 50!\), \(c = 100!\)
• Expressions to evaluate: \(\log_a c\), \(\log_c a\), \(\log_b a\), \(\log_a b\)

Let's compute each logarithm using the property \(\log_x y = \frac{\log y}{\log x}\):

1. \(\log_a c = \frac{\log(100!)}{\log(30!)}\)
   Since \(100! > 30!\), this is a large number.
2. \(\log_c a = \frac{\log(30!)}{\log(100!)}\)
   Since \(30! < 100!\), this is a small number.
3. \(\log_b a = \frac{\log(30!)}{\log(50!)}\)
   Since \(30! < 50!\), this is a relatively small number.
4. \(\log_a b = \frac{\log(50!)}{\log(30!)}\)
   Since \(50! > 30!\), this is a relatively large number.

Now, let's write out the approximate comparative sizes based on the factorial properties:

• \(\log_c a\) will be the smallest because \(\log(30!) \ll \log(100!)\).
• \(\log_b a\) will be larger than \(\log_c a\) but smaller than \(\log_a b\).
• \(\log_a b\) will be next since \(\log(50!) > \log(30!)\).
• \(\log_a c\) will be the largest because \(\log(100!) \gg \log(30!)\).

This order matches the first option provided:

\(\log_c a < \log_b a < \log_a b < \log_a c\)

Therefore, the correct inequality is:

\(\log_c a < \log_b a < \log_a b < \log_a c\)