Can anyone help me clear a whole forest?

I'm pretty sure that the correct syntax that you are looking for is "/fill air replace log". You put the log before the air but it is really the other way around. So once youve correctly used the command i listed above, you should just have to wait for the leaves to decompose and youre golden (:. Just be sure that you list the coordinates correctly.

From the command wiki:

fill <x1> <y1> <z1> <x2> <y2> <z2> <block> <dataValue|state> replace [replaceTileName] [replaceDataValue]

You have the fill block and the replace block backwards. Your command will try to replace all air with birch logs.

This command functions properly:

/fill ~-15 ~-10 ~-15 ~15 ~10 ~15 minecraft:air 0 replace minecraft:log 2

After which you can also use:

/fill ~-15 ~-10 ~-15 ~15 ~10 ~15 minecraft:air 0 replace minecraft:leaves 

This will replace any type of leaves so be careful if you are trying to save any other type of tree. If you are clearing any type of tree then you can omit the 2 and the end of the first command to clear any type of log as well.

These commands can be run each time you move around as they will clear a 30 X 20 X 30 area.

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If you are wanting to just run around to clear trees, you can put use the execute command in a repeating block:

execute @p ~ ~ ~ fill ~-5 ~ ~-5 ~5 ~10 ~5 minecraft:air 0 replace minecraft:log 2

You can add a chain block to clear the leaves as well. The coordinates are a bit more conservative to run smoothly.

Don't forget to turn it off when you are done.