Definition and Characteristics of the Natural Logarithm
The natural logarithm, denoted as ln, is a crucial function in mathematics with unique properties. It is defined as the inverse function of the exponential function e^x and is only defined for positive real numbers, specifically on the interval ]0; +∞[.
Definition: The fonction logarithme népérien (natural logarithm function) is the inverse of the exponential function e^x.
Highlight: A key characteristic of the natural logarithm is its domain, which is strictly positive real numbers, excluding zero.
The natural logarithm has a special relationship with its inverse, the exponential function. This relationship is expressed through two fundamental equations:
- For all real x, ln(e^x) = x
- For all x > 0, e^ln(x) = x
Example: If we take ln(e^2), the result is simply 2, as the natural logarithm "undoes" the exponential function.
These properties make the natural logarithm essential in solving exponential equations and in various fields of mathematics and science.