Consider the set


$$C=\left\{\log\_\frac{a+1}{b+1}\frac{a}{b} : a\ne b\in\mathbb{Z}^+\right\}$$
The set $C$ cannot contain all real numbers in $[1,\infty)$ because it is a countable set. But is it dense in $[1,\infty)$, or in some subinterval of it (of positive length)?


To prove that it is dense, we would need that for any $r\in[1,\infty)$ and $\epsilon > 0$, there exist $a,b$ such that
$$\left|r-\log\_\frac{a+1}{b+1}\frac{a}{b} \right| < \epsilon. $$
That is, we need $\log\_\frac{a+1}{b+1}\frac{a}{b}$ close to $r$. The form of the expression does not allow us to solve for $a,b$ though.
There was no answer when I [posted](https://math.stackexchange.com/questions/4115953/density-of-logarithm-with-fractional-base) on Stackexchange a while ago. I wonder what tools can be used to solve this problem.



Density of logarithm with fractional base
$\newcommand{\ep}{\epsilon}\newcommand{\N}{\mathbb N}$The answer is no. Indeed, suppose the contrary: that the set of all values of
\begin{equation\*}
 l(a,b):=\log\_\frac{a+1}{b+1}\frac ab \tag{0}
\end{equation\*}
for distinct natural numbers $a$ and $b$ is dense in the interval $[c,d]$ for some real $c$ and $d$ such that $c<d$ and
\begin{equation\*}
 c>1. \tag{1}
\end{equation\*}
Note that $l(b,a)=l(a,b)$.
So, the set
\begin{equation\*}
 P\_c:=\{(a,b)\in\N^2\colon b<a,\, l(a,b)\ge c\} \tag{2}
\end{equation\*}
is infinite.


**Lemma 1:** $l(a,b)\to1$ if $1\le b<a\to\infty$.


This lemma will be proved at the end of this answer.


At this point, note that, by Lemma 1, (1), and (2), there exists some real $A\_c>0$ depending only on $c$ such that $1\le b<a\le A\_c$ for all $(a,b)\in P\_c$. So, the set $P\_c$ is finite, a contradiction.


It remains to provide


*Proof of Lemma 1:* Suppose that $1\le b<a<\infty$. Then
\begin{equation}
 l(a,b)=\frac{\ln a-\ln b}{\ln(a+1)-\ln(b+1)}=\frac{\xi+1}\xi
\end{equation}
for some $\xi\in[b,a]$, by [Cauchy's mean value theorem](https://en.wikipedia.org/wiki/Mean_value_theorem#Cauchy%27s_mean_value_theorem). So, $l(a,b)\to1$ if $b\to\infty$. On the other hand, if $b\ge1$ stays bounded while $a\to\infty$, then
\begin{equation}
 l(a,b)\sim\frac{\ln a}{\ln(a+1)}\to1. 
\end{equation}
This completes the proof of Lemma 1 and thus the proof of the entire claim.
$\Box$


