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 \( y=x \).
\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
\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.
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