Exponential and logarithmic functions

Part 2 · Functions · Chapter 4 · lecture notes by Fabio Furini · Chapter PDF

1. Exponential functions

Definition 1: exponential functions

Given \(b \in \R_+\setminus \{1\}\), the function:

\[\begin{equation} \label{epeonziali} f: \mathbb{R} \rightarrow \mathbb{R},~~~ f:x \mapsto b^x \end{equation}\]

is called the exponential function with base \(b\).

• Regarding positivity and monotonicity, we have:

  \[ b^x> 0, ~~~\forall x \in \R {\rm ~~~and~~~} \begin{cases} {\rm ~~if~~} 0 < b < 1 {\rm ~~~then~} f {\rm ~~is~decreasing~on~} \R\\[3ex] {\rm ~~if~~} b > 1 {\rm ~~~then~} f {\rm ~~is~increasing~on~} \R \end{cases} \]

2. Logarithmic functions

Definition 2: logarithmic functions

Given \(a \in \R_{>0}\setminus \{1\}\), the function:

\[\begin{equation} \label{epeonziali__2} f: (0,+\infty) \rightarrow \mathbb{R},~~~ f:x \mapsto \log_a x \end{equation}\]

is called the logarithmic function with base \(a\).

• Regarding positivity and monotonicity, we have:

  \[ \begin{cases} {\rm ~~if~~} 0 < a < 1 & \log_a x > 0,~~ \forall x \in (0,1),~~\log_a x < 0,~~ \forall x \in (1,\ip){\rm ~~~and~~~} f {\rm ~~is~decreasing~on~} \R_{>0} \\[4ex] {\rm ~~if~~} a > 1 & \log_a x < 0,~~ \forall x \in (0,1),~~\log_a x > 0,~~ \forall x \in (1,\ip) {\rm ~~~and~~~} f {\rm ~~is~increasing~on~} \R_{>0} \end{cases} \]

  Moreover, we have:

  \[ \log_a x = 0 \Longleftrightarrow x=1,~~~ \forall a \in \R_{>0}\setminus \{1\} \]

Try it — the interactive graph below shows what you have just read: move the sliders.

3. Change of base

Given any base \(c \in \R_{>0}\setminus \{1\}\), all exponential and logarithmic functions can be rewritten in terms of another base \(d \in \R_{>0} \setminus \{1\}\) as follows:

\[ c^x = d^{\log_d c^x} = d^{x \; \log_d c} {\rm ~~~~~hence~with~~} \lambda = \log_d c {\rm ~~~~we~get~~~~~} c^x = d^{\lambda \; x} \]

\[ \log_c x = \frac{\log_d x}{\log_d c} {\rm ~~~~~hence~with~~} \lambda=\frac{1}{\log_d c} {\rm ~~~~we~get~~~~~} \log_c x= \lambda \; \log_d x \]

The logarithmic function with base \(e\) is also written \(\log x\) or \(\ln x\), while the one with base \(2\) is also written \(\lg x\).

Example 1: graphs of exponential and logarithmic functions

Exponential functions:

Logarithmic functions: