Chapter 4.3, Problem 98E

To determine

To find: The domain, vertical asymptote and x -intercepts of the function $f\left(x\right)={\log }_{3}x$ and draw its graph.

Answer to Problem 98E

The graph is shown below and thedomain, the vertical asymptote and the x -intercept are $\left(0,\infty \right)$ , $x=0$ and $\left(1,0\right)$ , respectively.

Explanation of Solution

Given information:

The function is $f\left(x\right)={\log }_{3}x$ .

Calculation:

Draw the graph of the function $f\left(x\right)={\log }_{3}x$ as shown below.

  

The function $f\left(x\right)={\log }_{3}x$ is real and defined for $x>0$ so the domain of the function will be $\left(0,\infty \right)$ .The line $x=L$ is a vertical asymptote of the function $y=f\left(x\right)$ , if the limit of the function (one-sided) at this point is infinite. The possible points are points where the denominator equals to 0 or doesn’t exist.

Let’s find the points where the denominator equals to 0 and check them.

Calculate the vertical asymptote.

  $\begin{array}{c}f\left(x\right)={\log }_{3}x\\ =\frac{\log x}{\log 3}\end{array}$

The denominator equals to 0 or does not exist at $x=0$ . So, the vertical asymptote is $x=0$ .

Substitute 0 for $f\left(x\right)$ in the function $f\left(x\right)={\log }_{3}x$ to find the x -intercepts.

  $\begin{array}{c}0={\log }_{3}x\\ x=3^0\\ =1\end{array}$

So, the x-intercept is $\left(1,0\right)$ .

The graph is shown above and thedomain, the vertical asymptote and the x -intercept are $\left(0,\infty \right)$ , $x=0$ and $\left(1,0\right)$ , respectively.