# Graphing Natural Logarithmic Functions

The inverse function of $y=e^x$ is $y=\log_{e}{x}$. Therefore $y=\log_{e}{x}$ is an inverse function, it is a reflection of $y=e^x$ in the line $y=x$.

The graphs of $y=e^x$ is $y=\log_{e}{x}$:

\begin{array}{|c|c|c|} \require{color} \hline
& y=e^x & \color{red}y =\log_{e}{x} \\ \hline
\text{domain} & x \in \mathbb{R} & \color{red}x \gt 0 \\ \hline
\text{range} & y \gt 0 & \color{red}y \in \mathbb{R} \\ \hline
\text{asymptote} & horizontal\ y=0 & \color{red}vertical\ x=0 \\ \hline
\text{fixed point} & (0,1) & \color{red}(1,0) \\ \hline
\end{array}

### Example 1

Sketch graphs of $y=\log_{e}{x}$ and $y=\log_{e}{(2x)}$.

### Example 2

Sketch graphs of $y=\log_{e}{x}$ and $y=2\log_{e}{x}$.

### Example 3

Sketch graphs of $y=\log_{e}{x}$ and $y=\log_{e}{(-x)}$.

Reflected by $y$-axis.

### Example 4

Sketch graphs of $y=\log_{e}{x}$ and $y=-\log_{e}{x}$.

Reflected by $x$-axis.

### Example 5

Sketch graphs of $y=\log_{e}{x}$ and $y=\log_{e}{(x+1)}$.

Shifting to the left side by $1$ unit.

### Example 6

Sketch graphs of $y=\log_{e}{x}$ and $y=\log_{e}{x}+1$.

Moving up by $1$ unit.