Consider the exponential and logarithmic functions with the base $e$ , $exp_e(x) = e^x$ for $x \in R$ and $log_e(x) = ln(x)$ for $x \in (0,\infty)$. These functions are sometimes referred to as the "natural logarithm" and "natural exponential" functions.
From the definition of the logarithm functions as inverses of the corresponding exponential function and the special features of these functions with the base $e$, the number $e$ is given special treatment.In this section we consider the characterization and definition of $e$ given in courses prior to calculus, namely $e \approx (1+ \frac 1 n)^n$ when $n>>0$. This definition is often motivated by the use of tables but is seldom visualized at this early level with either a graph or a mapping diagram. In this section the definition of $e$ is visualized both with graphs and mapping diagrams by using $n = 2^k$ to make approximations of $e$ more efficiently and more visually accessible.
From ELF.IDA we have that $exp_e$ and $log_e$ are increasing functions.Furthermore, the values of $exp_e(x)$ are between $2^x$ and $3^x$ and the values of $ln(x)$ are between $log_2(x)$ and $log_3(x)$.