The area of the region bounded by the straight lines $x = \frac{1}{2}$ and $x = 2$, and the curves given by the equations $y = \log_e x$ and $y = 2^x$ is\\
(A) $\frac{1}{\log_e 2}(4 + \sqrt{2}) - \frac{5}{2}\log_e 2 + \frac{3}{2}$\\
(B) $\frac{1}{\log_e 2}(4 - \sqrt{2}) - \frac{5}{2}\log_e 2$\\
(C) $\frac{1}{\log_e 2}(4 - \sqrt{2}) - \frac{5}{2}\log_e 2 + \frac{3}{2}$\\
(D) none of the above