TITLE: What is the difference between the three types of logarithms?
QUESTION [14 upvotes]: In complex analysis I came across three types of logarithms
 namely $\ln$, $\log$ and $\text{Log}$. What is the difference between the three?

REPLY [6 votes]: These are notations, sometimes ambiguous, to denote potentially different types of logarithms, which depend on the language of origin (Russian, German, French), see for instance where $\lg$ is used, including number theory, since sometimes $\log_2 x $ denotes the iterated logarithm: $\log{\log{ x }}$. 
Notation $\ln x$ (almost) unambiguously denotes the natural logarithm $\log_e x$  (latin: logarithmus naturalis), or logarithm in base $e$. In French, I used to believe the "n" stood for "népérien", from Neper or Napier.
The notation $\log x$ should be the adopted notation for the natural logarithm, and it is so in mathematics. However, it often represents the "most natural" depending on the field: I learned it as the base-$10$ logarithm ($\log_{10} x$) at school, and it is often used this way in engineering (for instance in the definition of decibels): 
And it may also represent a base-$2$ logarithm (binary logarithm) in binary calculus. The latter is sometimes denoted lg, ld (logarithmus dualis), or lb. 
$\operatorname{Log} x$ normally is  the principal value for a complex number, with imaginary part in the interval $(-\pi,\pi]$. But, while I learned at school $\log x$ for $\log_{10} x$, $\operatorname{Log} x$ was used (wrongly) to denote the natural logarithm. Such notations can be found for instance in Calcul differentiel et integral. Tome 1, 1998, N. Piskounov, page 58 sq.
The History of logarithms wikipedia page is worth reading, if only to learn the origin of the name (a number indicating a ratio: logos, proportion, and arithmos, number), and John Napier's book: Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms).