real analysis – Density of logarithm with fractional base

Consider the set

$$C=left{log_frac{a+1}{b+1}frac{a}{b} : ane binmathbb{Z}^+right}$$
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 $rin(1,infty)$ 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 on Stackexchange a while ago. I wonder what tools can be used to solve this problem.