The natural logarithm is the logarithm with the base *e*. The Scottish mathematician John Napier (1550-1617) invented the logarithm. Although he did not introduce the concept of the natural logarithm himself, the function is sometimes called the Napierian logarithm. The natural logarithm is used in numerous scientific and engineering applications.

John Napier developed the name "logarithm" as a combination of the Greek words *logos* and *arithmos*. The English translations are "ratio" and "numbers," respectively. Napier spent 20 years working on his theory of logarithms and published his work in the book *Mirifici Logarithmorum canonis descriptio* in 1614. The English translation of the title is *A Description of the Marvelous Rule of Logarithms*.

The natural logarithm is characterized as the logarithm of base *e*, which is sometimes called Napier's Constant. This number is also known as Euler's number. The letter "e" is used to honor Leonhard Euler (1707-1783) and was first used by Euler himself in a letter to Christian Goldbach in 1731.

The inverse of the natural exponential function, defined as f(x) = e^{x}, is the natural logarithmic function. This function is written as f(x) = ln(x). This same function can be written as f(x) = log_{e}(x), but the standard notation is f(x) = ln(x).

The domain of the natural logarithm is (0, infinity) and the range is (-infinity, infinity). The graph of this function is concave, facing downward. The function itself is increasing, continuous and one-to-one.

The natural logarithm of 1 is equal to 0. Assuming that a and b are positive numbers, then ln(a*b) is equal to ln(a) + ln(b) and ln(a/b) = ln(a) - ln(b). If a and b are positive numbers and n is a rational number, than ln(a^{n}) = n*ln(a). These properties of natural logarithms are characteristic of all logarithmic functions.

The actual definition of the natural logarithmic function can be found in the integral of 1/t dt. The integral is from 1 to x with x > 0. Euler's number, *e*, denotes the positive real number such that the integral of 1/t dt from 1 to *e* is equal to 1. Euler's number is an irrational number and is approximately equal to 2.7182818285.

The derivative of the natural logarithmic function with respect to x is 1/x. The derivative with respect to x of the inverse of the logarithmic function, the natural exponential function, is surprisingly the natural exponential function again. In other words, the natural exponential function is its own derivative.