Question

If $\log _ { 3 } 2 , \log _ { 3 } \left( 2 ^ { x } - 5 \right)$ and $\log _ { 3 } \left( 2 ^ { x } - \frac { 7 } { 2 } \right)$ are in A.P., then $x$ is equal to.

• A.
• B.
• C. $1 , \frac { 3 } { 2 }$
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

$\log _ { 3 } 2 , \log _ { 3 } \left( 2 ^ { x } - 5 \right)$ and $\log _ { 3 } \left( 2 ^ { x } - \frac { 7 } { 2 } \right)$ are in A.P.

$\Rightarrow$ $2 \log _ { 3 } \left( 2 ^ { x } - 5 \right) = \log _ { 3 } \left[ ( 2 ) \left( 2 ^ { x } - \frac { 7 } { 2 } \right) \right]$

$\Rightarrow$ $\left( 2 ^ { x } - 5 \right) ^ { 2 } = 2 ^ { x + 1 } - 7$ $\Rightarrow$ $2 ^ { 2 x } - 12 \cdot 2 ^ { x } - 32 = 0$

$\Rightarrow$ $x = 2,3$

But $x = 2$ does not hold, hence $x = 3$ .