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  4. If log102, log10 (2x - 1) and log10(2x + 3) be three consecutive terms of an A.P. then

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Q. If $\log_{10}2, \log_{10} (2^x - 1)$ and $\log_{10}(2^x + 3)$ be three consecutive terms of an A.P. then

Sequences and Series

Solution:

By the given condition
$ 2\, log_{10} \left(2^{x}-1\right) = log_{10}\,2 + log_{10}\left(2^{x}+3\right)$
$ \Rightarrow \left(2^{x}-1\right)^{2} = 2\left(2^{x}+3\right) $
$ \Rightarrow 2^{2x} -2.2^{x} +1 = 2 \,2^{x} +6$
$ \Rightarrow 2^{2x}-4.2^{x}-5 = 0 $
$\Rightarrow \left(2^{x}-5\right)\left(2^{x}+1\right) = 0 $
$ 2^{x}= 5 \,or \,2^{x} = -1 $
But $2^{x} = -1$ is not possible.
$ \therefore 2^{x} = 5$
$ \Rightarrow log_{2} 2^{x} = log_{2} 5$
$\Rightarrow x log_{2} 2 = log_{2}5 $
$ \Rightarrow x= log_{2} 5 \left[\because log_{2} 2 = 1\right]$