I should define what I mean by two sequences $(a_n)$ and $(b_n)$ that are close together. What I mean is that for any $\epsilon > 0$, $\exists N$ such that for $n \geq N$ we have $|a_n-b_n| < \epsilon$. I will write $a_n \to b_n$ in this case. Neither sequence need converge. Now suppose that $\ln(a_n) \to \ln(b_n)$ for sequences that don't break the logarithm. Does $a_n \to b_n$?
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