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QJEEM21Ma803e: JEE Main 2021 Mathematics - Class 11 JEEM21Ma803e Solution: If are in an arithmetic progression, then the value of x is…

If log⁡32,log⁡3(2x−5),log⁡3(2x−72){\log _3}2,{\log _3}({2^x} - 5),{\log _3}\left( {{2^x} - {7 \over 2}} \right) are in an arithmetic progression, then the value of x is equal to _____________.

Step-by-Step Verified Solution

2log⁡3(2x−5)=log⁡2+log⁡3(2x−72)2{\log _3}({2^x} - 5) = {\log _2} + {\log _3}\left( {{2^x} - {7 \over 2}} \right)

Let 2x=t{2^x} = t

log⁡3(t−5)2=log⁡32(t−72){\log _3}{(t - 5)^2} = {\log _3}2\left( {t - {7 \over 2}} \right)

(t−5)2=2t−7{(t - 5)^2} = 2t - 7

t2−12t+32=0{t^2} - 12t + 32 = 0

(t−4)(t−8)=0(t - 4)(t - 8) = 0

⇒\Rightarrow 2x = 4 or 2x = 8

x = 2 (Rejected)

Or x = 3

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