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.The discrete estimation based definition of $e$ allows further argument about the nature of these functions from a rather elementary view in applications of these functions to various continuous growth and decay models and to continuous compound interest, so that it is often treated in algebra courses at the intermediate algebra level. Approaches to characterizing these special exponential functions can be quite varied after the concepts of calculus has been established. Definitions can use differentiation, integration, and infinite series.

From the estimates of $e$ we can understand that $ 2< 2.7 < e < 3$.

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 $log_3(x)$.These facts about the natural exponential and logarithm functions are often visualized in graphs. In the next examples these facts are also visualized using mapping diagrams.