Normal to a curve at point #P# lying on this curve, by definition, is a line perpendicular to a tangent to a curve at this point (presuming the curve is smooth and has a tangent, otherwise a normal is undefined).
Since #OP# is a tangent to curve #y=log_e(x)# from origin #O# to point #P# lying on this curve, it is also a perpendicular to a normal to a curve at point #P#.
So, our task is to measure the length of #OP#.
This is done by Pythagorean Theorem as the distance between two points:
origin #O# with coordinates #(0,0)# and
point #P# on a curve with coordinates #(X_P,Y_P)#, where
#Y_P = log_e(X_P)#.
#OP = sqrt(X_P^2+Y_P^2)=sqrt(X_P^2+log_e^2(X_P))#
