lf d is the minimum distance between the curves $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{\mathrm{x}} \mathrm{and} \mathrm{g}\left(\mathrm{x}\right)={\log }_{\mathrm{e}}\mathrm{x},$ then the value of a⁶ is_____

Since the graphs $\mathrm{y}={\mathrm{e}}^{\mathrm{x}} \mathrm{and} \mathrm{y} = \log \mathrm{x}$ are symmetrical about the line y = x, minimum distance is the distance along the common normal to both the curves, i.e., y = x must be parallel to the tangent as both the curves are inverse of each other.

$\begin{array}{l} {\left.\frac{\mathrm{dy}}{\mathrm{dx}}\right|}_{{\mathrm{x}}_1}={\mathrm{e}}^{{\mathrm{x}}_1}=1\\ \mathrm{or} {\mathrm{x}}_1=0 \mathrm{and} {\mathrm{y}}_1=1\\ \mathrm{or} \mathrm{A}\equiv \left(0,1\right) \mathrm{and} \mathrm{B}\equiv \left(1,0\right)\\ \mathrm{or} \mathrm{AB}=\sqrt{2}\end{array}$