How to find the solutions of the following equation?

Question

Let we have the following equation with the unknown $x$ $$\lfloor\ln(x+1)\rfloor - \lfloor\ln(x)\rfloor = 1$$ Where $\lfloor x\rfloor$ means the integer part of $x$

Answer 1

The solution set is a sequence of intervals. For each positive integer $N$, we can construct an interval containing $e^N$. Let

$$\epsilon = \ln\left(\frac{e^N}{e^N+1}\right).$$

Then the interval from $e^{N-\epsilon}$ to $e^N$ is a set of solutions.

You need

$$\ln(e^{N-\epsilon}+1)\geq N$$

and solving this inequality gives the value for $\epsilon.$