Question

Let \(a = 30!\), \(b = 50!\), and \(c = 100!\). Consider the following numbers: \(\log_a c\), \(\log_c a\), \(\log_b a\), \(\log_a b\)
Which one of the following inequalities is CORRECT?

• A. \(\log_a c<\log_a b<\log_b a<\log_a c\)
• B. \(\log_c a<\log_a b<\log_b a<\log_b c\)
• C. \(\log_c a<\log_a b<\log_a c<\log_b a\)
• D. \(\log_b a<\log_c a<\log_a b<\log_a c\)

Hint:

In logarithms, when comparing values of large factorials, smaller bases yield smaller logarithms. For factorials like \(a = 30!\), \(b = 50!\), and \(c = 100!\), \(\log_a c\) will be the smallest.

Answer 1

The Correct Option is A

Step 1: Understanding the values of \(a\), \(b\), and \(c\).
- \(a = 30!\), a very large number.
- \(b = 50!\), a larger number than \(a\).
- \(c = 100!\), which is even larger than \(b\).
Step 2: Evaluate logarithmic relations.
- \(\log_a c\) measures how many times you need to multiply \(a\) to get \(c\), which will be small, since \(c = 100!\) is much larger than \(a = 30!\).
- \(\log_c a\) is the inverse, and it will be much smaller than \(\log_a c\).
- \(\log_b a\) and \(\log_a b\) reflect the relationship between \(b = 50!\) and \(a = 30!\). Since \(b\) is larger, \(\log_a b\) will be larger than \(\log_b a\).
Step 3: Compare the inequalities.
- \(\log_a c\) is the smallest.
- \(\log_a b\) is the next in order.
- \(\log_b a\) is larger than both.
- \(\log_a c\) is the largest.
Thus, the correct inequality is: \[ \log_a c<\log_a b<\log_b a<\log_a c \] \[ \boxed{\text{The correct inequality is (A).}} \]