Amazon has "print to kindle" function that can be used to export to kindle rather conveniently.
The installation file for PC is here http://www.amazon.com/gp/sendtokindle/pc
For Macs http://www.amazon.com/gp/sendtokindle/mac
Although the printer appears in the list from the print/export dialog in Logos, I was not able to send directly to kindle from there. What worked for me was the 2 steps method: (1) From the Logos print/export pane: save as file (doc, docx and rtf are supported). (2) Right click on the saved file and click on "send to kindle". In the dialog that appears, pick the device where you want it sent (if you have more than one). Done.
I have not had the chance to test extensively. If someone finds a way to print directly from the Logos print/export, I'm sure many would like to know.