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))
