Exponential Functions

An exponential function has the form $f(x) = a^{x}$, where $a > 0$, $a \neq 1$. Any exponential function of this form has four traits you should know.

1. An exponential function passes through the point $(0,1)$, that is $a^{0}=1$.
2. An exponential function has a horizontal asymptote at $y=0$.
3. The domain of an exponential function is $(-\infty,\infty) = \left\{ x \mid x \in \mathbb{R} \right\}$.
4. The range of an exponential function is $(0,\infty) = \left\{ y \mid y>0 \right\}$.

The Natural Exponential Function

$\text{Euler's constant}$, denoted by the letter $\textrm{e}$, is defined as the irrational number satisfying the following limit (you will learn about limits in your calculus course): $$ \lim \limits_{n \rightarrow \infty} \left( 1 + \frac{1}{n} \right)^{n} = \textrm{e} \approx 2.71828\ldots.$$ Using Euler's constant, we can define an exponential function $f(x) = \textrm{e}^{x}$ called the $\text{natural exponential function}$. The function $f(x)=\textrm{e}^{x}$ satisfies all four traits of an exponential function.

1. The natural exponential function passes through the point $(0,1)$, that is $\textrm{e}^{0}=1$.
2. The natural exponential function has a horizontal asymptote at $y=0$.
3. The domain of the natural exponential function is $(-\infty,\infty) = \left\{ x \mid x \in \mathbb{R} \right\}$.
4. The range of the natural exponential function is $(0,\infty) = \left\{ y \mid y>0 \right\}$.

Algebraically, the natural exponential function $\textrm{e}^{x}$ satisfies all power rules.

1. $\textrm{e}^{0}=1$
2. $\textrm{e}^{1}=\textrm{e}$
3. $\displaystyle \textrm{e}^{-m} = \frac{1}{\textrm{e}^{m}}$
4. $\textrm{e}^{m}\textrm{e}^{n} = \textrm{e}^{m+n}$
5. $\displaystyle \frac{\textrm{e}^{m}}{\textrm{e}^{n}} = \textrm{e}^{m-n}$
6. $(\textrm{e}^{m})^{n} = \textrm{e}^{mn}$

Logarithmic Functions

A logarithmic function has the form $f(x) = \log_{a}(x)$, where $a > 1$. Any logarithmic function of this form has four traits you should know.

1. A logarithmic function passes through the point $(1,0)$, that is $\log_{a}(1)=0$.
2. A logarithmic function has a vertical asymptote at $x=0$.
3. The domain of a logarithmic function is $(0,\infty) = \left\{ x \mid x>0\right\}$.
4. The range of a logarithmic function is $(-\infty,\infty) = \left\{ y \mid y \in \mathbb{R}\right\}$.

The Natural Logarithmic Function

Since $\textrm{e}$ is a positive real number not equal to one, we can define the logarithmic function $f(x) = \log_{\textrm{e}}(x) = \ln(x)$, called the natural logarithmic function. The function $f(x)=\ln(x)$ satisfies all four traits of a logarithmic function.

1. The natural logarithmic function passes through the point $(1,0)$, that is $\ln(1)=0$.
2. The natural logarithmic function has a vertical asymptote at $x=0$.
3. The domain of the natural logarithmic function is $(0,\infty) = \left\{ x \mid x>0\right\}$.
4. The range of the natural logarithmic function is $(-\infty,\infty) = \left\{ y \mid y\in \mathbb{R} \right\}$.

The natural logarithmic function $\ln(x)$ satisfies all logarithm laws.

1. $\ln(pq) = \ln(p) + \ln(q)$
2. $\displaystyle \ln \left(\frac{p}{q}\right) = \ln(p) - \ln(q)$
3. $\ln(p^{k}) = k \ln(p)$

Finally, the natural logarithm and the natural exponential are inverse functions of each other. This leads to the following important identities:

1. $\ln(\textrm{e}^{x})=x$, $x \in \mathbb{R}$
2. $\textrm{e}^{\ln(x)} = x$, $x>0$
3. $\ln(1) = 0$
4. $\ln(\textrm{e}) = 1$