Question

If log₁₀2, log₁₀(2ˣ - 1) and log₁₀(2ˣ + 3) are three consecutive terms of an arithmetic progression, then
(a). x = 0.
(b). x = 1.
(c). x = log₂5
(d). x = log₁₀2

{AMU +2 ENTRANCE TEST 2016-17}

Answer 1

hello dear hope its help you

Answer 2

log₁₀2, log₁₀(2ˣ - 1) and log₁₀(2ˣ + 3) are I. AP

therefore,

2 log₁₀(2ˣ - 1) = log₁₀2+log₁₀(2ˣ + 3)

log₁₀(2ˣ - 1)² = log₁₀2(2ˣ + 3)

=> (2ˣ - 1)² = 2(2ˣ + 3)

=> 2²ˣ-2.2ˣ+1 = 2.2ˣ+6

=> 2²ˣ-4.2ˣ-5 = 0

=> 2²ˣ-5.2ˣ+1.2ˣ-5 = 0

=> 2ˣ(2ˣ-5) + 1(2ˣ-5) = 0

=> (2ˣ+1)(2ˣ-5)=0

Since (2ˣ+1)=0 is not possible
{ 2ˣ=-1 // not possible}

=> (2ˣ-5)=0

=> x = log₂5