How to get rid of the ceiling function on one side of an equation

Question

Given this function,

$\left\lceil \log _{b-1} \left(\lceil \frac{1}{b} \rceil\right) \right\rceil=1$

How do I get rid of the ceiling function so I can obtain the value of $B$ ?

Answer 1

By definition, $x\le \lceil x\rceil<x+1$; therefore $\lceil y\rceil = N$ is equivalent to $N-1< y\le N$.

Answer 2

If $b \lt 0$ the ceiling of $\frac 1b$ is at most $0$ and the log is undefined. If $b \ge 1$ the ceiling of $\frac 1b$ is $1$ and the log is $0$, which fails, so we have $0 \lt b \lt 1$. In that case $b-1 \lt 0$. The definition I know for logs to a base different from $e$ is $\log_a(x)=\frac {\log x}{\log a}$. If $b-1 \lt 0, \log_{b-1}x$ is undefined and there is no answer.