If log3 2, log3 (2^x - 5), log3 (2^x - 7/2) are in arithmetic progression

• Progressions
• Logarithms

If log₃ 2, log₃ (2^x - 5), log₃ (2^x - 7/2) are in arithmetic progression, then the value of x is equal to

1. 2
2. 3
3. 4
4. 5

Answer

For three numbers in arithmetic progression,

2 log₃ (2^x - 5) = log₃ 2 + log₃ (2^x - 7/2)

log₃ (2^x - 5)² = log₃ (2)(2^x - 7/2)

(2^x - 5)² = 2(2^x - 7/2)

Let 2x = y

(y - 5)² = 2(y - 7/2)

y² + 25 - 10y = 2y - 7

y² - 12y + 32 = 0

(y - 8)(y - 4) = 0

y = 4 or 8

x = 2 or 3

But, if x= 2 then 2nd term is negative, so x = 3

The correct option is B.