Chapter 8.4, Problem 36PPS

(a)

To determine

To Find: The shapes of the graphs compared and determine how the asymptotes and the x intercept of the graph compare.

Answer to Problem 36PPS

The shape of the graph is same and the asymptote is the y axis and the x intercept is 1.

Explanation of Solution

Given:

The given diagram is shown in Figure 1

  

Figure 1

The given log function are ${\log }_{4}x$ and ${\log }_{\frac{1}{4}}x$ .

Calculation:

The shape of the graph shows that the they are same and the asymptote of the graph is the y axis and the x intercept of each graph is 1.

(b)

To determine

To Find: The relationship between the graphs.

Answer to Problem 36PPS

The relation between the graphs is that they are the reflections of each over the x axis.

Explanation of Solution

The relation between the graph is that they are the reflections of each over the x axis.

(c)

To determine

To Find: The comparison contrast the graph of each of the function and the graph of $y={\log }_{4}x$ with the graph of the given equation.

Answer to Problem 36PPS

The comparison contrast the graph of each of the function and the graph of $y={\log }_{4}x$ with the graph of the given equation is obtained.

Explanation of Solution

Given:

The given logarithm is,

  $\begin{array}{l}y={\log }_{4}x+2\\ y={\log }_4\left(x+2\right)\\ y=3{\log }_{4}x\end{array}$

Calculation:

The graph for the three function and the graph for $y={\log }_{4}x$ is shown in Figure 1

  

Figure 1

The graph for $y={\log }_{4}x+2$ and its graph is shifted 2 units to the left of graph for $y={\log }_{4}x$ .

  $\begin{array}{l}y={\log }_4\left(x+2\right)\\ y=3{\log }_{4}x\end{array}$

The graph for $y={\log }_4\left(x+2\right)$ and its graph is shifted 2 units to the upward of graph for $y={\log }_{4}x$ .

The graph for $y=3{\log }_{4}x$ and its graph is shifted 3 times of graph for $y={\log }_{4}x$ .

(d)

To determine

To Find: The relationship between the $y={\log }_{4}x$ and $y=-1{\log }_{4}x$ , then find the reasonable domain and the range of the function.

Answer to Problem 36PPS

The graph are the reflection of each other across the x axis.

Explanation of Solution

Consider the function is $y={\log }_{4}x$ and $y=-1{\log }_{4}x$ . Both the graph are the reflection of each other across the x axis.

(e)

To determine

To Find: The equation for the function for which the graph is the graph for the $y={\log }_{3}4$ translated 4 unit to the left and the 1 unit up.

Answer to Problem 36PPS

The equation is $y={\log }_3\left(x+4\right)+1$ .

Explanation of Solution

Consider the standard equation of the graph is,

  $f\left(x\right)=a\log \left(x-h\right)+k$

Since, there is the horizontal shift of 4 units and vertical shift of the 1 unit. Then,

  $f\left(x\right)={\log }_3\left(x+4\right)+1$