If you complete Spider's Nest (Requires Level 32 Carpenter) you will be able to store logs and other resources at Storage Rifts, making training easier).
Cut the logs at the Circular Saw at the South end of the workshop.
If the logs have a final product, use them on the Lathe to the East of the Saw.
Now you can train by purchasing logs from Gwen's Lumber.
You will need about 20 Silver and an empty inventory to start buying logs.
Buy 12 Logs at a time, and process them. If you have the Silver, you can save time by amassing a larger amount of Logs at the Lumber Bank, then take out 12 Logs at a time.
Keep upgrading your logs every two levels
Optional: Skip logs like Juniper that don't have a final product to save money
Tips
At lower levels, you can sustain your Log purchasing costs by converting your knowledge into Silver.
The following page can be used as a reference to calculate Carpenter EXP rates hourly.