I haven't found anything that matches your quote, but there are lots of songs about log cabins!

The following are in the DT:
An Old Log Cabin For Sale
The Little Log Cabin by the Stream

And these are in the forum:
Cabin on the Hill
Little Log Cabin by the Sea
Little Log Cabin by the Sea
Little Old Log Cabin in the Lane

The Library of Congress American Memory Collection has links to sheet music for these old songs, and more:
Little Old Log Cabin in the Lane
The Little Log Cabin on the Hill
The Old Log Cabin in the Dell
The Old Log Cabin on the Hill