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{AMSsymbols} \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

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. 


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