The GATE Grind

GATE 2023 CE (CE1) – Question 9

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

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?

  1. $\log_c a < \log_b a < \log_a b < \log_a c$
  2. $\log_c a < \log_a b < \log_b a < \log_b c$
  3. $\log_c a < \log_b a < \log_a c < \log_a b$
  4. $\log_b a < \log_c a < \log_a b < \log_a c$

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Correct answer: (A) $\log_c a < \log_b a < \log_a b < \log_a c$

Explanation

Since $a < b < c$ and all are greater than 1, $\log_a b > 1$ and $\log_a c > \log_a b > 1$, and $\log_b a$ and $\log_c a$ are less than 1. Also $\log_b a = \frac{1}{\log_a b}$ and $\log_c a = \frac{1}{\log_a c}$, so the larger base gives the smaller value: $\log_c a < \log_b a$. The order is $\log_c a < \log_b a < \log_a b < \log_a c$.