Solving Logarithms involving ceiling function

Question

I need to solve the equation

$\lceil \log_B(M) \rceil = S$

for $B$ when $M$ and $S$ are known, $M$ and $S$ are integers, and $B < M$. Were the ceiling function not there, it would be trivial, but it is there, and I don't know how to do algebra with ceiling functions

Answer 1

$$S=\lceil \log_{B}M\rceil$$

So we need $B^{S-1}<M\le B^S$ since the logarithm is an increasing function. This gives us $B<M^{\frac{1}{S-1}}$ and $B\ge M^{\frac{1}{S}}$. Hence any $B\in [M^{\frac{1}{S}},M^{\frac{1}{S-1}})$ works.