The area bounded by the curve $y={\log }_{e}x$, $X-$ axis  and the ordinate $x=e$ is 

The given curve is $y={\log }_{e}x$

The rough sketch of the required area as shown below

The required area is the area of shaded reigon, this is the area of the region bounded by the curve $y=\log x$ and the lines $x=0,x=1$ and $x-$ axis

Hence,  the required area is $A={\int }_1^e{\log }_{e}xdx$

use the integration by parts, to evaluate the above integral

hence, 

 $\begin{array}{rcl}A& =& {\int }_1^{e}1·{\log }_{e}xdx\\ & =& x·\log x-{\int }_1^{e}1dx\\ & =& e-\left(e-1\right)\\ & =& 1\end{array}$

Therefore, the required area is 1 square unit.