Graphing Logarithmic Functions

Graph of Logarithmic Function with a>1

Here is what the graph of a logarithmic function $f\left(x\right)=\log_ax,\:\text{for}\:a>1\:$looks like, and a list of characteristics from this graph.

Characteristics of Graph of Logarithmic Function with a>1

• Domain: $\left(0,\infty\right)$
• Range: $\left(-\infty,\infty\right)$
• $f\left(x\right)=\log_ax,\:\text{for}\:a>1\:$is increasing and continuous on its entire domain.
• The y-axis is a vertical asymptote as $x\longrightarrow0\:$from the right.
• The graph passes through the points $\left(\frac{1}{a},-1\right),\left(1,0\right),\:\text{and}\:\left(a,1\right).$

Example

Here is a Khan video showing how to graph $y=\log_5x$.

Caution:

If you write a logarithmic function in exponential form to graph, as in the Video Example above, start first with y-values to calculate corresponding x-values. Be careful to write the values in the ordered pairs in the correct order.

Graph of Logarithmic Function with 0<a<1

Here is what the graph of a logarithmic function $f\left(x\right)=\log_ax,$ $\:\text{for}\:0<a<1\:$looks like, and a list of characteristics from this graph.

Characteristics of Graph of Logarithmic Function with 0<a<1

• Domain: $\left(0,\infty\right)$
• Range: $\left(-\infty,\infty\right)$
• $f\left(x\right)=\log_ax,\:\text{for}\:0<a<1\:$is decreasing and continuous on its entire domain.
• The y-axis is a vertical asymptote as $x\longrightarrow0\:$from the right.
• The graph passes through the points  $\left(\frac{1}{a},-1\right),\left(1,0\right),\:\text{and}\:\left(a,1\right).$