Graphing Logarithmic Functions

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 functions, we can see that we have a common logarithm and natural logarithm. Recall that the base of the common logarithm is $10,$ and the natural logarithm has the number $e$ as its base. With this in mind, let's highlight the base of each function. $$\begin{array}{ccc}f(x)={\log }_{2}x& \iff & f(x)={\log }_{2}x\\ g(x)=\ln x& \iff & g(x)={\log }_{e}x\\ h(x)=\log x& \iff & h(x)={\log }_{10}x\end{array}$$

Determining Points on the Graph

Using the base $b$ we can identify three points on the graph of a logarithmic function. $$\begin{array}{c}\left(\frac{1}{b},\text{-}1\right),(1,0),\text{and}(b,1)\end{array}$$ Since we know the bases, we can immediately determine these points.

Plotting the Points and Sketch the Graph

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

We can see that, for the three functions, the $x\text{-}$variable takes values greater than $0,$ and the $y\text{-}$values are all real numbers. Therefore, the three functions have the same domain and range. $$\begin{array}{rl}\text{Domain:}& x>0\\ \text{Range:}& \text{All real numbers}\end{array}$$ We can also see that, for the three functions, their values approach infinity as $x$ approaches infinity, and their values approach negative infinity as $x$ approaches $0$ on the positive side. The line $x=0$ is the vertical asymptote of the three graphs. None of them have a horizontal asymptote. $$\begin{array}{cc}\text{Vertical}& \text{Horizontal}\\ \text{asymptote}& \text{asymptote}\\ x=0& \text{none}\end{array}$$ The $x\text{-}$intercept is the same for the three graphs, and it occurs at $(1,0).$ None of the graphs intercept the $y\text{-}$axis, as it is a vertical asymtpote. The three graphs increase in their entire domain, and are negative for values of $x$ greater than $0$ and less than $1.$ Lastly, the three graphs are positive for $x\text{-}$values greater than $1.$ $$\begin{array}{rl}\mathbf{x}\text{-intercepts:}& 1\\ \mathbf{y}\text{-intercepts:}& \text{none}\\ \text{Increasing Interval:}& x>0\\ \text{Decreasing Interval:}& \varnothing \\ \text{Positve Set:}& x>1\\ \text{Negative Set:}& 0<x<1\end{array}$$