You can get good function generators for < $200. Have a look at the FY6900, Uni-T UTG962 or the Juntek PSG9080. They have been widely discussed on this forum.
They all have the capability to perform automatic sweeps in either linear or log fashion. All of them use a rotary encoder to change values such as frequency, phase offset, etc. So you could perform a manual linear sweep with the rotary encoder. A manual log sweep is not directly supported. However, all of these units can be controlled from a computer (or microcontroller over a serial interface) so such a capability could be implemented with a little programming work on your part.
Are you only interested in audio frequency wave forms? Perhaps you can just use your computer's sound card. There are a lot of programs to do this -- just do a web search for "sound card function generator".