Given a flat object in Illustrator, how can I scale it along the X or Y axis so that it stretches in an exponential fashion? That means that the center barely is stretched, but the outer extents stretch much more? I know there are some warp effects, but those seem to affect both axes.
Scaling the object using guides:
I made this template which contains the exponential/logarithmic lines.
Create a new letter size Document in Illustrator and Place the template.
Because in this case I’m going to use the exponential lines, we are going to lock it and hide the logarithmic lines.
Import your flat object into Illustrator.
Select your object and go to Object > Envelope Distort > Make with Mesh.
Depending on the orientation of the scaling, insert 13 and 1 respectively.
Now, that you applied the Envelope Distort to the object, Align each row/column of anchors with the template.