What is the difference between the three types of logarithms? In complex analysis I came across three types of logarithms
 namely $\ln$, $\log$ and $\text{Log}$. What is the difference between the three?
ln is always a natural logarithm (of base $e$).
log if it has a base like $\log_5x$ then the base is the specified one, otherwise it is either base $e$ or base $10$. It depends on people. Some people (mostly high school people) use logarithms without a base as base $10$ and other people as base $e$. It is most often used as a natural logarithm though but you have to watch out. If you see a ln button on a calculator, then log without base is guaranteed to be base 10 on the same calculator.


(Edit: As @ClementC. pointed out, in computer science, $log$ without a base most often means a binary logarithm.)


Log stands for complex logarithm in its principal branch. The principal branch has its imaginary part in the interval $(−π,π]$. (Basically it's the inverse of the complex exponential function with its imaginary part in that interval.)
I thing if $z=re^{i\theta}$ then
$\log z=\ln r+i\theta$. Now if in the arguement $\theta ,n=1$ then "$\log$" becomes "Log". Also "$\ln$" is generally used with real part of complex number as per the notations used by C.V.Chirchill