First, let's start with logarithms:
In the early 17th century John Napier and Joost Burgi independently discovered logarithms. Logarithms made complicated calculations much easier to compute (Rooney, 40). The natural logarithm is just a logarithm with the base of e. It is called natural because e appears over and over again in mathematics and log_e provides a simple way to find solutions to problems involving e^x.

Now, a little bit about the number e:
In 1689, Jacob Bernoulli was working on calculating compound interest. Specifically, he was looking at interest that was compounded continuously, so instead of interest being compounded once a month or once a week or even once a day, he wanted to look at interest that had no breaks between the times it was compounded. In essence, he was looking at the limit as x goes to infinity of (1+1/n)ⁿ. When evaluated, the value of the limit gives a value that is approximately 2.71828 which we call e (Elwes, 237).

And finally, we take a look at log_e, or the natural log:
By 1640 mathematicians knew how to calculate the area under curves that have the form yⁿ=ax^m. The only exception to this was the curve a/x, which is the standard form of the equation for the graph of a hyperbola (Flashman). Gregorius Saint-Vincent published a book in 1647 called Opus Geometricum, in which he found the area under a hyperbola using the sums of rectangles in a geometric progression. This method was very closely related to logarithms, but Saint-Vincent didn't make the connection between his method and logarithms. Instead, one of his students, Alphonse Antonio de Sarasa, was the one that made the connection between Saint-Vincent's work and logarithms (O'Connor).
The logarithm that corresponds to this work is the logarithm with base e, which was commonly called the hyperbolic logarithm, since it gave the area under the hyperbola. Nicolas Mercator was the first to use the term natural logarithm to denote log_e, but this wasn't until about 1668 (Anderson, 236). The first published work containing the computations of several hyperbolic logarithms was published in 1667 by James Gregory (Gonzalez). Now, the natural logarithm is denoted by ln and is used widely throughout mathematics and in many other fields, although now we have calculators and computers to do all of the hard work of computing a logarithm for us.