# 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 $y=\log_{2}{(x+1)}$.

Shift to the left side by $1$ unit.

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

Shift down by $2$ units.

(c)   Sketch the graphs of $y=\log_{2}{x}$ and $y=\log_{2}{(x+1)-2}$.

Shift to the left side by $1$ and down by $2$ units.

(d)   Find the domain of $y = \log_{2}{(x+1)}-2$.

\begin{align} \displaystyle x + 1 &\gt 0 \\ \therefore x &\gt -1 \\ \end{align}

(e)   Find the range of $y = \log_{2}{(x+1)}-2$.

$x \in \mathbb{R}$ or all real $x$

(f)   Find any asymptote(s) of $y = \log_{2}{(x+1)}-2$.

\begin{align} \displaystyle x + 1 &= 0 \\ \therefore x &= -1 \\ \end{align}

(g)   Find any $x$-intercept(s) of $y = \log_{2}{(x+1)}-2$.

\begin{align} \displaystyle \require{color} \log_{2}{(x+1)}-2 &= 0 &\color{red}y=0 \\ \log_{2}{(x+1)} &= 2 \\ x+1 &= 2^2 \\ x+1 &= 4 \\ x &= 3 \\ \therefore (3,0) \\ \end{align}

(h)   Find any $y$-intercept(s) of $y = \log_{2}{(x+1)}-2$.

\begin{align} \displaystyle y &= \log_{2}{(0+1)}-2 &\color{red}x=0 \\ y &= \log_{2}{1}-2 \\ y &= 0-2 \\ y &= -2 \\ \therefore (0,-2) \\ \end{align}

### 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 $y=\log_{2}{(-x)}$.