Graphing Logarithmic Functions

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 ${\color{#FF0000}{b}}={\color{#FF0000}{5}}.$ $\begin{gathered} f(x)=\log_{\color{#FF0000}{5}} (x) \end{gathered}$

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 ${\color{#FF0000}{b}}={\color{#FF0000}{5}},$ we can immediately identify these points as follows. $\begin{gathered} (1,0), \quad ({\color{#FF0000}{5}},1), \quad \text{and}\quad \left( \tfrac{1}{{\color{#FF0000}{5}}}, \text{-} 1 \right) \end{gathered}$

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 $0<b<1,$ the curve decreases. Try it below!

$\mathbf{b > 1}$

$\mathbf{0 < b < 1}$

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