First slide

Arithmetic progression

Question

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

Moderate

A

no real $x$
B

exactly one real $x$
C

exactly two real $x$
D

more than two real $x$

Solution

$2, 2^{x} - 1, 2^{x} + 3$ are in G.P.
$\begin{array}{l} \Rightarrow {(2^{x} - 1)}^{2} = 2 (2^{x} + 3) \\ \Rightarrow 2^{2 x} - 2 (2^{x}) + 1 = 2 (2^{x}) + 6 \\ \Rightarrow 2^{2 x} - 4 (2^{x}) - 5 = 0 \\ \Rightarrow (2^{x} - 5) (2 x + 1) = 0 \\ \Rightarrow 2^{x} = 5 [∵ 2^{x} + 1 > 0] \end{array}$
Thus, there is exactly one value of $x .$

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