The natural logarithm is really an area under a curve
Draw the curve y equals 1 over x and shade the region beneath it, starting at 1 and stopping at some number a. That shaded area is the natural logarithm of a. The definition is so clean that it earned the word natural, and it is how seventeenth-century mathematicians first stumbled on the function.
Put more familiarly, the natural logarithm of a number is the power you must raise e to in order to get that number. The constant e is irrational and roughly 2.718. So ln e is 1, ln 1 is 0, and ln 7.5 comes out a little over 2. For numbers between 0 and 1 the shaded area is counted as negative, so their logarithms are negative too, sinking toward minus infinity as the input approaches zero.
Its most useful trick is turning multiplication into addition: the log of a product equals the sum of the logs, and the log of a power becomes a simple multiple. That is why logarithms are the tool for equations where the unknown sits in an exponent, such as working out half-lives in radioactive decay or how long compound interest takes to grow a sum. Logarithms in any other base are just the natural one scaled by a constant.
The history runs through the hyperbola. Before 1649, Grégoire de Saint-Vincent and Alphonse Antonio de Sarasa, studying the area under the curve xy equals 1, found that it behaved like what they called a hyperbolic logarithm. Nicholas Mercator discussed it in his 1668 Logarithmotechnia and gave his name to a series that expresses ln of 1 plus x as an endless alternating sum. Earlier still, in 1619, the teacher John Speidell had produced a table that effectively listed natural logarithms, though stored as whole numbers.
Notation varies by field, which catches people out. Mathematicians and many programming languages often write log x and mean the natural logarithm, as in the prime number theorem. Chemists usually mean base 10 by the same symbols, and computer scientists discussing running times often mean base 2. The function also links to the harmonic series: adding 1 plus a half plus a third and so on up to N gives roughly ln N, the gap settling on a fixed constant.
Source: Natural logarithm
