IPMAT Indore 2025 · Quantitative Ability · Question 20

Work from the outside in, peeling off one logarithm at a time.

Step 1 — undo the outer {\log}_{25}. If {\log}_{25}T=1/2, then T={25}^{1/2}=5:

5{\log}_{3}\left( 1+{\log}_{3}\left( 1+2{\log}_{2}x \right) \right)=5

Step 2 — divide by 5.

{\log}_{3}\left( 1+{\log}_{3}\left( 1+2{\log}_{2}x \right) \right)=1

Step 3 — undo that {\log}_{3} (if {\log}_{3}T=1 then T=3):

1+{\log}_{3}\left( 1+2{\log}_{2}x \right)=3 \Rightarrow{\log}_{3}\left( 1+2{\log}_{2}x \right)=2

Step 4 — undo the next {\log}_{3} (now T={3}^{2}=9):

1+2{\log}_{2}x=9 \Rightarrow{\log}_{2}x=4

Step 5 — undo the last log.

x={2}^{4}=16

Answer: Option 2 — 16