HMMT 2005 U22

Find

$$\{\ln (1+e)\}+\left\{\ln \left(1+e^{2}\right)\right\}+\left\{\ln \left(1+e^{4}\right)\right\}+\left\{\ln \left(1+e^{8}\right)\right\}+\cdots,$$

where $\{x\}=x-\lfloor x\rfloor$ denotes the fractional part of $x$.