GATE Civil 2023 Set 1 | GA Question: 9

Question

Let $a=30 !, b=50!,$ and $c=100!.$ Consider the following numbers:

$$\log _a c, \quad \log _c a, \quad \log _b a, \quad \log _a b$$

Which one of the following inequalities is $\text{CORRECT?}$

• A. $\log _c a<\log _b a<\log _a b<\log _a c$
• B. $\log _c a<\log _a b<\log _b a<\log _b c$
• C. $\log _c a<\log _b a<\log _a c<\log _a b$
• D. $\log _b a<\log _c a<\log _a b<\log _a c$

Answer 1

Answer- Option A

 

If we want to compare values of 2 log functions, they need to have the same base.

Every given log value contains $a$, either in base or in argument.Hence, we can represent all values such that they have the same base $a$.

 

$log_ac = log_{30!}100!$

$log_ca = \frac{1}{log_ac} = \frac{1}{log_{30!}100!}$

$log_ab = log_{30!}50!$

$log_ba = \frac{1}{log_ab} = \frac{1}{log_{30!}50!}$

 

If $x_{1} > x_{2}$,  $log_ax_{1} > log_ax_{2}$ where $a > 1$

Hence,

$ log_{30!}100! > log_{30!}50! > \frac{1}{log_{30!}50!} > \frac{1}{log_{30!}100!}$

 

$log_ac > log_ab > log_ba > log_ca$

This is same as option A.

Answer 2

A is the answer. You can approximate the the values like 30! = 30^n and then find log.