# 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.