How to find the solutions of the following equation?

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 $$x$$

• Use formula for difference of logs. – coffeemath Jan 1 '19 at 10:09
• Why did you enclose in square parentheses [ ] each one of those expressions? Why not merely write $\;\ln(x+1) - \ln x\;$ ? Does this mean anything special? And after this, what properties of logarithms you know? – DonAntonio Jan 1 '19 at 10:09
• It is [x] integer part of x, I think – Aqua Jan 1 '19 at 10:11
• Nice. The OP hasn't even addressed the comment, but someone already decided that he can read minds and edited the question... – DonAntonio Jan 1 '19 at 11:22
• but there is two answers already. – onepound Jan 1 '19 at 13:55

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.
$$\ln(e^{N-\epsilon}+1)\geq N$$
and solving this inequality gives the value for $$\epsilon.$$