Hi. I have a way that hopefully clears things up.
First, we set (i) 10x=e^bx because this is the point at which the two lines, y=10x and y=e^bx, will intersect in the xy-plane.
I solved for 10, and got (ii) 10=(e^bx)/x.
The derivative of y=e^bx is y'=be^bx. The derivative will equal 10 at a certain point, thus (iii)10=be^bx.
I set the two 10's equal to each other, thus (iv) (e^bx)/x=b(e^bx). If you solve for b on the right hand side, you get (v) b=1/x.
Plug (1/x) in for b in equation (i) and we find that x=e/10.
By subbing in e/10 for x in equation (v) it's clear that b=10/e.