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# Graphing Logarithmic Functions

## Graphing Logarithmic Functions 1.6 - Solution

To draw the graph of a logarithmic function, we can follow a three-step process.

1. Identify the base.
2. Determine points on the graph.
3. Plot the points and sketch the graph.

Let's do it!

### Identifying the Base

Looking at the given function, we can see that the base is ${\color{#FF0000}{b}}={\color{#FF0000}{\frac{1}{3}}}.$ $\begin{gathered} f(x)=\log_{\color{#FF0000}{{\frac{1}{3}}}} (x) \end{gathered}$

### Determining Points on the Graph

Using the base, we can identify three points on the graph of a logarithmic function. $\begin{gathered} (1,0), \quad ({\color{#FF0000}{b}},1), \quad \text{and}\quad \left( \tfrac{1}{{\color{#FF0000}{b}}}, \text{-} 1 \right) \end{gathered}$ Since we know that ${\color{#FF0000}{b}}={\color{#FF0000}{\frac{1}{3}}},$ we already have that $({\color{#FF0000}{b}},1)$ $=$ $\left({\color{#FF0000}{\frac{1}{3}}},1 \right).$ Let's now calculate $\left( \frac{1}{{\color{#FF0000}{b}}}, \text{-} 1 \right).$
$\left(\dfrac{1}{{\color{#FF0000}{b}}},\text{-}1\right)$
$\left(\dfrac{1}{{\color{#FF0000}{1/3}}},\text{-}1\right)$
Simplify
$\left(\dfrac{1(3)}{1},\text{-}1\right)$
$\left(\dfrac{3}{1},\text{-}1\right)$
$\left(3,\text{-}1\right)$

### Plotting the Points and Sketch the Graph

Finally, we will plot the three points and connect them with a smooth curve.

### Extra

Useful Theory

The graph depends on the value of the base $b.$ If $b>1,$ the curve increases. If $0 the curve decreases. Try it below!

$\mathbf{b > 1}$

$\mathbf{0 < b < 1}$

Reset