The natural exponential function and logarithm

Introduction to differentiation: Derivatives of exponential functions and logarithms

The natural exponential function and logarithm

An exponential function is the representation of an exponential growth process. Its derivative shows the growth of that process. A characteristic feature of exponential growth is:

Exponential growth

We say that a quantity grows exponentially if the growth rate at any moment is proportional to the value of the quantity at that time.

If a quantity at time $t$ is given by a nonzero constant multiple of an exponential function: $f(t)=b \cdot a^t$ , for given real numbers $a$ and $b$ with $a\gt0$ , then we have:

$f'(t)=c \cdot f(t)\tiny,$ where $c$ is a real number that satisfies:

$\begin{array}{rcl}c\lt0&\text{ for }&0 \lt a \lt 1\\ c=0&\text{ for }& a=1\\ c\gt0 &\text{ for }& a\gt1\end{array}$

In particular, $f$ then grows exponentially.

The theorem is true for all real (including negative) values of $t$ . This property can be understood by looking at the derivative of the exponential function $f(t)=b\cdot a^{t}$ . To this end, we first determine the difference quotient of $f$ in $t$ with difference $h$ :

$\begin{array}{rcl} \dfrac{\Delta f}{\Delta t} &=& \dfrac{f(t+h)-f(t)}{h} \\ &&\phantom{x}\color{blue}{\text{definition of }\Delta}\\ &=& \dfrac{b\cdot a^{t+h}-b\cdot a^{t}}{h} \\&&\phantom{x}\color{blue}{\text{function rules }f(t)\text{ and }f(t+h) \text{ substituted}}\\ &=& b\cdot a^t \cdot \dfrac{ a^h-1}{h}\\ &&\phantom{x}\color{blue}{b\cdot a^t \text{ placed in front of fraction}}\\\end{array}$ For $h \to 0$ , this difference quotient becomes the derivative of $f$ in $t$ :

$\begin{array}{rcl}f'(t)&=&\lim_{h \to 0}\left(b\cdot a^t \cdot \dfrac{a^h-1}{h}\right)\\&=&b\cdot a^t \cdot\lim_{h \to 0}\frac{a^h-1}{h}\\ &=& f(t)\cdot c\\&=& c\cdot f(t)\end{array}$ where $c=\lim_{h \to 0}\dfrac{a^h-1}{h}$ is a number that only depends on $a$ . This indeed shows that $f'(t)=c \cdot f(t)$ . The value for $c$ is given by $c=\lim_{h \to 0}\dfrac{a^h-1}{h}=g'(0)$ , where $g(t)=a^t$ .

For $0\lt a\lt 1$ the function $g$ is decreasing so $g'(0)\lt0$ thus $c\lt0$ .
For $a=1$ the function $g$ is constant, so $g'(0)=0$ thus $c=0$ .
For $a\gt1$ the function $g$ is increasing, so $g'(0)\gt0$ thus $c\gt0$ .

This proves the last part of the statement.

Since

$c$ increases if $a$ increases,
$c\lt1$ if $a=2$ and
$c\gt1$ if $a=3$ ,

we expect there to be a base number between $2$ and $3$ for which the constant $c$ is exactly $1$ . This number indeed exists:

Euler's number

There exists a number $\e$ such that $\lim_{h \to 0}\dfrac{\e^h-1}{h}=1$ . It is called Euler's number and denoted as $\e$ . It is a real number and is approximated by

$\e\approx 2.71828182846\tiny.$

The proof can be given by first establishing that $c$ , as a function of $a$ , is continuous and then applying the mean value theorem.

Natural exponential function and logarithmic function

By $\exp$ we denote the exponential function: $\exp(x) = {\e}^x$ .

By $\ln$ we denote the inverse function of $\exp$ .

The function $\exp$ is also called the natural exponential function and $\ln$ the natural logarithm.

If $a$ is a positive real number, then the function is $a^x$ equals $\e^{\ln(a)\cdot x}$ . The range of this function is $\ivoo{0}{\infty}$ and its inverse is

$\log_a(x)=\frac{\ln(x)}{\ln(a)}\tiny.$

The function $\log_a$ is called the log to the base $a$ .

The function $\exp$ is strictly increasing and its range is $\ivoo{0}{\infty}$ . Therefore, this function is injective: any positive real number is the value $\exp(x)$ of exactly one $x$ . From this it follows that the inverse function of $\exp$ is defined on the domain $\ivoo{0}{\infty}$ .

The equality $a^x=\e^{\ln(a)\cdot x}$ follows from

$\e^{\ln(a)\cdot x}=\left(\e^{\ln(a)}\right)^{x}=a^x\tiny.$

The fact that $\log_a(x)$ is the inverse of $a^x$ follows from

$a^{\log_a(x)}=a^{\frac{\ln(x)}{\ln(a)}}=\e^{\ln(a)\cdot \frac{\ln(x)}{\ln(a)}}=\e^{\ln(x)}=x\tiny.$

Here is the graph of the function $\log_a$ . The values of $a$ can be varied using the slider.