Natural Logarithm

natural logarithm (ln), logarithm with base e = 2.718281828…. That is, ln (e^x) = x, where e^x is the exponential function. The natural logarithm function is defined by ln x = Integral on the interval [1, x ] of ∫ 1 x ^dt/_tfor x > 0; therefore the derivative of the natural logarithm is^d/_dx ln x = ¹/_x. The natural logarithm is one of the most useful functions in mathematics, with applications throughout the physical and biological sciences.

The natural logarithm follows the same rules as the common logarithm (logarithm with base 10, usually written as log). That is, ln (ab) = ln a + ln b; ln (a/b) = ln a – ln b; and ln (a^b) = b ln a. The natural logarithm and the common logarithm are related throughln x = ^{log x}/_{log e}log x = ^{ln x}/_{ln 10}.