# Graphing Logarithmic Functions

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

The graphs of $y=a^x$ is $y=\log_{a}{x}$ for $0 \lt a \lt 1$: The graphs of $y=a^x$ is $y=\log_{a}{x}$ for $a \gt 1$: \begin{array}{|c|c|c|} \require{color} \hline & y=a^x & \color{red}y =\log_{a}{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

Consider the function $y = \log_{2}{(x+1)}-2$.

(a) Sketch the graphs of $y=\log_{2}{x}$ and $y=\log_{2}{(x+1)}$.
(b) Sketch the graphs of $y=\log_{2}{x}$ and $y=\log_{2}{x-2}$.
(c) Sketch the graphs of $y=\log_{2}{x}$ and $y=\log_{2}{(x+1)-2}$.
(d) Find the domain of $y = \log_{2}{(x+1)}-2$.
(e) Find the range of $y = \log_{2}{(x+1)}-2$.
(f) Find any asymptote(s) of $y = \log_{2}{(x+1)}-2$.
(g) Find any $x$-intercept(s) of $y = \log_{2}{(x+1)}-2$.
(h) Find any $y$-intercept(s) of $y = \log_{2}{(x+1)}-2$.

### Example 2

Sketch the graphs of $y=\log_{2}{x}$ and $y=3\log_{2}{x}$.

### Example 3

Sketch the graphs of $y=\log_{2}{x}$ and $y=\log_{2}{(3x)}$.

### Example 4

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

### Example 5

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