The area of the region bounded by $y = \log x, y = 2$ and the coordinate axes is

Required area = area of rectangle $OABC –$ area under $y = \log x, x = 1, x = e^2$
    $\begin{array}{l}= AO\times OC-{\int }_1^{e^2} \log xdx\\ = 2e^2-[x\log x-x{]}_1^2\\ = 2e^2-\left[2e^2-e^2\right]-1=e^2-1.\end{array}$