If $$\log 2, \log(2^{x}-1)$$ and $$\log(2^{x}+3)$$ are in A.P, then x is equal to ______________

If $$\log 2, \log(2^{x}-1)$$ and $$\log(2^{x}+3)$$ are in A.P

We know when A, B and C are in AP.

2B = A+C

Thus, $$2\log(2^x-1)\ =\log2+\log\left(2^x+3\right)$$

Assuming $$2^x=p$$

$$2\log(p-1)\ =\log2+\log\left(p+3\right)$$

Now, applying the properties of logarithms

$$\log_{a}{xy} = \log_{a}{x}+\log_{a}{y}$$

$$(p-1)^2\ =2\left(p+3\right)$$

$$\ p^2-2p+1=2p+6$$

$$\ p^2-4p-5=0$$

$$\left(p-5\right)\left(p+1\right)=0$$

Thus, p = 5 or -1

$$2^x$$ = 5 as it cannot be negative.

Thus, x = $$\log_25$$