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CAT 2022 Slot 2 — QA Question 1

LogarithmsEasy

The number of distinct integer values of n satisfying 4-log2n3-log4n < 0, is

Answer & solution

Answer: 47

Solution

Easy

A fraction is negative exactly when its numerator and denominator have opposite signs. Split into two cases and count the integers nn that satisfy each. Remember the domain: a logarithm needs n>0n>0.

1

Case 1: numerator <0<0, denominator >0>0.

\begin{aligned} &4-\log_2 n<0 \;\Rightarrow\; \log_2 n>4 \;\Rightarrow\; n>16\\ &3-\log_4 n>0 \;\Rightarrow\; \log_4 n<3 \;\Rightarrow\; n<4^3=64\\ &\Rightarrow\ 16Integers 17,18,…,6317,18,\dots,63, that is 63−17+1=4763-17+1=47 values.

2

Case 2: numerator >0>0, denominator <0<0.

4−log⁡2n>0  ⇒  n<163−log⁡4n<0  ⇒  n>64\begin{aligned} &4-\log_2 n>0 \;\Rightarrow\; n<16\\ &3-\log_4 n<0 \;\Rightarrow\; n>64 \end{aligned}

No nn can be both <16<16 and >64>64, so this case gives 00 values.

47+0=4747+0=\mathbf{47} distinct integers.

Once you see the only feasible case is $16