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The natural logarithm, invented by John Napier, is the logarithm to the base e, also known as Napier’s constant or Euler’s number. Its inverse is the natural logarithmic function, written as ln(x), with properties such as ln(a*b) = ln(a) + ln(b). Its derivative is 1/x, and the natural exponential function is its own derivative.
The natural logarithm is the logarithm to the base e. 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 Napier’s logarithm. The natural logarithm is used in a number of 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 to the base e, 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) = ex, is the natural logarithmic function. This function is written as f(x) = ln(x). This function itself can be written as f(x) = loge(x), but the standard notation is f(x) = ln(x).
The domain of the natural logarithm is (0, infinity) and the interval is (-infinity, infinity). The graph of this function is concave, pointing downwards. The function itself is increasing, continuous, and bijective.
The natural logarithm of 1 equals 0. Assuming a and b are positive numbers, then ln(a*b) equals 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, then ln(an) = n*ln(a). These properties of natural logarithms are characteristic of all logarithmic functions.
The actual definition of the natural logarithmic function is found in the integral of 1/t dt. The integral runs from 1 to x with x > 0. The Euler number, e, denotes the positive real number such that the integral of 1/t dt from 1 to e equals 1. The Euler 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 again the natural exponential function. In other words, the natural exponential function is its own derivative.
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