Can you prove or disprove this floor/ceiling equation for positive integers and , ?

I think I can prove the equation when either or is a positive integer power of as follows.

When for positive integer , the original equation becomes . Which can be proven because .

Similarly, when for positive integer , the original equation becomes . Like before, this is easy to prove because .

Is this correct so far? I'm not sure what implications this has for when neither nor is a positive integer power of . Any pointers you can offer will be greatly appreciated. Thanks!