Exponential curve to follow path

Hello,

I am in the process of building a loudspeaker enclosure which requires modelling a folded horn. Please check the image for visual representation.

So i've been playing with equation driven curves etc but can't figure out a elegant way to create an exponential curve which follows a path. Check other upload for clarity.

Will I have to plot this out manually?

Any tips would be very much appreciated! 

Thanks,
Craig

SolidworksSketching