If log 2, log(2^x-1) and log(2^x+3) are in A.P. ,then the value of x is ?

Since the terms are in AP

So , 2*log(2^x - 1) = log2 + log(2^x + 3)

 

log(2^x - 1)² = log(2(2^x + 3))

 

Let 2^x = t

 

So , (t-1)² = 2(t+3)

t² + 1 - 2t = 2t + 6

t² - 4t - 5 = 0

(t -5)(t + 1) = 0

t = 5 , -1

But -1 is not as 2^x cannot be -1 for any value of x

So 2^x  = 5 ,

x = log₂5