Natural Logarithm
The natural logarithm, also known as the Napierian logarithm, is a logarithm with base equal to Euler's number (e). It is usually written as ln or sometimes as log without an explicit base. $$ \ln = \log = \log_e $$ where e=2.71828 is Euler's number.
The natural logarithm is the inverse of the exponential function ex.
$$ \ln x = f^{-1}(e^x) $$
This means that applying the exponential function and the natural logarithm one after the other returns the original value.
$$ e^{\ln(x)}=x $$
Which symbol is used for the natural logarithm?
The notation used for natural logarithms is not always the same across different books and disciplines.
In many mathematics textbooks, the natural logarithm is written using the symbol ln.
$$ \ln x $$
It may also appear in the equivalent form
$$ \log_e x $$
where the base e is written explicitly.
However, not all authors follow the same convention.
In some texts, especially in applied sciences, the symbol log without a specified base is used to denote the natural logarithm.
$$ \log x $$
For this reason, it is always important to check how the notation log x is defined in a particular book or article.
Note. In many mathematics textbooks, log(x) denotes the natural logarithm. In engineering and scientific applications, however, log(x) often refers to the common logarithm, that is, the base-10 logarithm log10(x). In these notes, the notation log without an explicit base refers to the natural logarithm, unless otherwise stated.
The History of Natural Logarithms
Natural logarithms were not originally introduced by Napier ( John Napier ).
Although the Scottish mathematician introduced logarithms and discovered Euler's number e=2.71828 in the 16th century, he did not use it as the base of logarithms.
It was the mathematician Euler ( Leonhard Euler ) in the 18th century who adopted Euler's number as the base of natural logarithms, particularly in the study of powers with imaginary exponents.
Euler also introduced the symbol "e" to represent Euler's number.
And so on.
