Chapter 6 Elementary Functions

Section 6.4 Exponential and Logarithmic Functions

6.4.3 The Natural Exponential Function

There is a very special exponential function, sometimes also called the exponential function, that we will study now. In fact, all other exponential functions can be reduced to this special exponential function. It has Euler's number $e$ as its base. Its value is (approximately) equal to

${\displaystyle e=2.718281828459045235\dots \hspace{0.5em}.}$


So, let us consider the graph of the exponential function - for the time being without any additional parameters -

${\displaystyle g:\mathrm{\hspace{0.5em}\hspace{0.5em}}\{\begin{array}{ccc}\hfill ℝ& \hfill \to \hfill & (0;\infty )\hfill \\ \hfill x& \hfill ⟼\hfill & g(x)=e{}^x\hfill \end{array}}$


which is, because of its base $e$, also called the $e$ function or natural exponential function: Unsurprisingly, the natural exponential function shows the typical behaviour of exponential functions $x\mapsto a^x$ $(a>1)$ already discussed in Section 6.4.2, after all we have only chosen a special value for the base, namely $a=e$. In particular, we note again that the natural exponential function is strictly increasing, i.e. for large negative values of $x$, it approaches the negative $x$-axis, and for $x=0$, it takes the value $1$.

At the beginning of this subsection we claimed that the exponential functions described above can be reduced to the natural exponential function. This is done by means of the identity

${\displaystyle a^x=e{}^{x·\ln (a)}}$

that is valid for any real number $a>0$ and any real number $x$. Here, $\ln$ denotes the natural logarithmic function that will be studied in detail in the following Section 6.4.4.
In general natural exponential functions, the parameters $f_0$ and $\lambda$ occur that were already introduced in Section 6.4.2; thus, its functional description is as follows:

${\displaystyle f:\mathrm{\hspace{0.5em}\hspace{0.5em}}\{\begin{array}{ccc}\hfill ℝ& \hfill \to \hfill & (0;\infty )\hfill \\ \hfill x& \hfill ⟼\hfill & f(x)=f_0·e{}^{\lambda x}\hfill \end{array}\hspace{0.5em}.}$


Again, the parameter $f_0$ describes initial values different from $1$, and the factor $\lambda$ in the exponent allows for different (positive or negative) growth rates. This shall be finally illustrated by means of an example.