I am wondering if the following expectation can be computed in the shown manner
$$\begin{aligned} E[\log_2(1+X) \mid X>a] &= \int_{0}^{\infty} P( \log_2(1+X)>t \mid X>a) dt \\ &= \int_{0}^{\infty} P(X>2^t-1 \mid X>a) dt \\ &= \int_{0}^{\infty} \frac{P(X>2^t-1 \cap X>a)}{P(X>a)} dt \\ &= \frac{1}{P(X>a)} \left(\int_{0}^{a} P(X>a) dt + \int_a^{\infty} P(X>2^{t}-1) dt \right) \end{aligned}$$
Note that there is no closed form expression for the PDF of X hence I am trying to find alternative ways to compute the expectation.
Thank you