Introduction 1: The exponential function with base $e$, $y(x)=e^x$, is frequently used in economic growth models. The number $e$ ($\approx 2.718$) is called the number of Eule}.
Introduction 2: A function of the form $y(x)=\;^a\!\log x, (x>0)$ where $a$ ($a\neq 1$) is a positive number is called a logarithmic function with base $a$.
Definition: The logarithm with base $e$ is called the natural logarithm and is denoted as \[ y(x)=\ln x. \]
Remark: Note that $\textrm{ln}(x)=\;^e\!\log x$.
The graph of the natural logarithm $y(x)=\textrm{ln}(x)$ is shown in the following figure.
Introduction 2: A function of the form $y(x)=\;^a\!\log x, (x>0)$ where $a$ ($a\neq 1$) is a positive number is called a logarithmic function with base $a$.
Definition: The logarithm with base $e$ is called the natural logarithm and is denoted as \[ y(x)=\ln x. \]
Remark: Note that $\textrm{ln}(x)=\;^e\!\log x$.
The graph of the natural logarithm $y(x)=\textrm{ln}(x)$ is shown in the following figure.