Is it correct to use 'Ln' instead of 'ln' for natural logarithm? In some research papers, authors use 'Ln' for natural logarithm instead of 'ln'.  Is it correct? 
 A: The forms $\ln$ or $\log_\mathrm{e}$ are standard in all fields I'm familiar with.† Though $\mathrm{Ln}$ might've become a standard, it didn't; & though it's unlikely to cause more than a momentary hesitation on the reader's part, even that is worth taking pains to avert. Moreover, I've noticed that its use is correlated with the commission of graver mathematical solecisms, so you may want to eschew it to avoid making a bad impression among those that know and care about such things.
In business notation is often sloppy, & while you're likely to see $\mathrm{Ln}$ often enough, it doesn't constitute an alternative standard, or necessarily result from a deliberate choice—could well be due to Powerpoint's autocorrect feature.
† Complex analysis isn't one of them—see @Zen's answer.
A: Why wouldn't it be? As long as it isn't confusing. 
However, it is sometimes the case (especially when writing in non-mathematical fields) that there is confusion over base 10 vs. base e, and often people in those fields haven't heard of (or have forgotten about) the difference between ln (or Ln) and log. 
If there is possibility of confusion, it is best to specify the base, and do so explicitly: $\log_{10}$ or $\log_\mathrm{e}$ or $\ln$ (base e logarithm) or something similar.
A: My guess is they were worried that in whatever typeface the paper was published in, the ln would look too much like the word In. I saw a typescript (dating myself) where ℓn [that is Latex \ell] was used, which meant changing the type ball of an IBM Selectric.
A: Be careful, because the notation $\mathrm{Ln}$ is currently used in mathematics. For $z\in\mathbb{C}$, the complex logarithm is the multivalued function defined by
$$
  \mathrm{Ln}(z) = \ln(|z|)+i(\arg(z)+2k\pi) 
$$
for $k=0,\pm 1,\pm 2, \dots$. Hence, you should check the mentioned papers to verify in which context they use the $\mathrm{Ln}$ notation.
Take a look.  
Also, there is some multiplicity of notations. For instance, some people write $\ln(z)$ for the complex logarithm of $z$ and use $\mathrm{Ln}(z)$ to denote its principal value (the value with $k=0$). The notations $\mathrm{Log}(z)$ and $\log(z)$ are also commonly used with the same mixed meanings.
