If 3^a=4, 4^b=5, 5^c=6, 6^d=7, 7^e=8, 8^f=9 find abcdef:

Question

UPSC Quizroom · Accepted Answer

We are given:

3^a = 4 → a = log₃4
4^b = 5 → b = log₄5
5^c = 6 → c = log₅6
6^d = 7 → d = log₆7
7^e = 8 → e = log₇8
8^f = 9 → f = log₈9

We need to find abcdef.

Step 1: Express using logarithms

abcdef = (log₃4) × (log₄5) × (log₅6) × (log₆7) × (log₇8) × (log₈9)

Change of base formula: log_a b = ln b / ln a

abcdef = (ln4 / ln3) × (ln5 / ln4) × (ln6 / ln5) × (ln7 / ln6) × (ln8 / ln7) × (ln9 / ln8)

Step 2: Simplify the telescoping product

abcdef = ln9 / ln3 = log₃9 = 2

Answer: 2

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