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