It is not your axe
This is the single most-asked question in the community. The fallen tree at Howling Marshes cannot be chopped regardless of tool level. Access is unlocked by story progression, a pattern the developers have confirmed applies to several blocked areas and systems.
What to do instead
Keep advancing main story quests and character quests rather than grinding tool upgrades for this specific blocker. Quest-gating in Moonlight Peaks also covers other forms and areas, so a blocked route usually means an undiscovered quest, not a missing tool.