### 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#.

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.

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