I am looking for formulas similar to classical logarithmic identities: $$a^{\log_ab}=b$$ $$a^{\log_cb}=b^{\log_ca}$$
involving floor or ceiling functions.
I need to evaluate the following expressions:
$$a^{\lfloor \log_ab \rfloor}$$ $$a^{\lceil \log_ab \rceil}$$
Is it possible to assert something about the last two expressions?
As the comment say the inequalities based on monotonicity seems the best we can conclude. We have $$\log_ab-1<\lfloor\log_ab\rfloor\le\log_ab,$$ and since the function $x\mapsto\log_ax$ is continuous this inequality cannot be sharpened. Indeed if $b=a^n$ for some integer $n$ we have the equality on the right, and $b$ can be arbitrarily close to some integer power of $a$ from the left so that $\lfloor\log_ab\rfloor$ is arbitrarily close to the lower bound. Again since the function $x\mapsto a^x$ is continuous the best we can conclude is $$b/a<\lfloor\log_ab\rfloor\le b$$ (if $a>1$) or $$b\le\lfloor\log_ab\rfloor<b/a$$ (if $0<a<1$).