Chapter 3, Problem 9CT

To determine

Find the domain, $x$ -intercept and vertical asymptote for the given logarithmic function and draw a graph.

Answer to Problem 9CT

Domain $=x>0$ .

There is no $x$ -intercept.

Vertical asymptote is $x=0$ .

Explanation of Solution

Given information:

The given functionis $f\left(x\right)=-\log x-6$ .

Calculation:

Domain $\to$ it is the set of input values for which the function is real and defined.

The function gives the positive values, when $x>0$ .

Domain $=x>0$ .

  $x$ -intercept $\to$ it is the point, where $y=0$ .

  $\begin{array}{l}f\left(x\right)=-\log x-6\\ \Rightarrow 0=-\log x-6\\ \Rightarrow 0+6=-\log x-6+6\\ \Rightarrow 6=-\log x\\ \Rightarrow -6=\log x\\ \Rightarrow {10}^{-6}=x\end{array}$

The given function is not defined at ${10}^{-6}$ .

There is no x-intercept.

Vertical asymptote $\to$ the logarithmic function $f\left(x\right)=c{\log }_a\left(x+h\right)+k$ has a vertical asymptote $x=-h$ .

In the function $f\left(x\right)=-\log x-6$ ,

  $h=0$

Vertical asymptote is $x=0$ .

Create a table list for some values for the function.

$x$	$f\left(x\right)=-\log x-6$
$\begin{array}{l}0.1\\ 5\\ 25\\ 35\end{array}$	$\begin{array}{l}-5\\ -6.699\\ -7.398\\ -7.544\end{array}$

The graph of the function is given below.