If ${\log _3}2,{\log _3}({2^x} - 5),{\log _3}\left( {{2^x} - {7 \over 2}} \right)$ are in an arithmetic progression, then the value of x is equal to _____________.
Answer (integer)
3
Solution
$$2{\log _3}({2^x} - 5) = {\log _2} + {\log _3}\left( {{2^x} - {7 \over 2}} \right)$$<br><br>Let ${2^x} = t$<br><br>${\log _3}{(t - 5)^2} = {\log _3}2\left( {t - {7 \over 2}} \right)$<br><br>${(t - 5)^2} = 2t - 7$<br><br>${t^2} - 12t + 32 = 0$<br><br>$(t - 4)(t - 8) = 0$<br><br>$\Rightarrow$ 2<sup>x</sup> = 4 or 2<sup>x</sup> = 8<br><br>x = 2 (Rejected)<br><br>Or x = 3
About this question
Subject: Mathematics · Chapter: Sequences and Series · Topic: Arithmetic Progression
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