Chapter 0.5, Problem 58E

a.

To determine

To graph: The function $y_1\text{ and }{y}_2$ for $a=2,3,4$ , and 5 and explain the relation between $y_1\text{ and }{y}_2$ .

Answer to Problem 58E

The graph of $y_1$ is defined as,

  

And, the graph of $y_2$ is defined as,

  

The function become equal when the value of $a$ is 1.

Explanation of Solution

Given Information:

The function is defined as,

  $y_1=\ln \left(\frac{x}{a}\right),y_2=\ln x$

Consider the given function,

  $y_1=\ln \left(\frac{x}{a}\right),y_2=\ln x$

Now, draw the graph by using the graphing calculator for the different value of $a$ .

  

Now, draw the second function by using the graphing calculator.

  

It can observed that as the value of $a$ is increasing then the value of $y_1$ is decreasing and $y_2$ is the parent function of $y_1$ . For $a=1$ the value of both function will be equal.

b.

To determine

To graph: The function $y_3$ for the value of $a=2,3,4,5$ .

Answer to Problem 58E

The graph of the function $y_3$ is defined as,

  

And the function is constant function.

Explanation of Solution

Given information:

The function is defined as,

  $y_1=\ln \left(\frac{x}{a}\right),y_2=\ln x$

And,

  $y_3=y_2-y_1$

Expalnation:

Consider the given functions,

  $y_1=\ln \left(\frac{x}{a}\right),y_2=\ln x$

Find the function $y_3$ by using the above functions.

  $\begin{array}{c}{y}_3=y_2-y_1\\ =\ln x-\ln \left(\frac{x}{a}\right)\\ =\ln \frac{x}{\frac{x}{a}}\\ =\ln a\end{array}$

Now, write the function for the value of $a=2,3,4,5$ .

  $\begin{array}{l}{y}_3=\ln 2\left[a=2\right]\\ y_3=\ln 3\left[a=3\right]\\ y_3=\ln 4\left[a=4\right]\\ y_3=\ln 5\left[a=5\right]\end{array}$

Draw the graph of the model by using the above points.

  

As it can be observed in the graph, the functions are the constant function.

Therefore, the function $y_3$ is the constant function.

c.

To determine

To graph: The function $y_4$ for the value of $a=2,3,4,5$ .

Answer to Problem 58E

The graph of the function $y_4$ is defined as,

  

And the function is constant function.

Explanation of Solution

Given information:

The function is defined as,

  $y_1=\ln \left(\frac{x}{a}\right),y_2=\ln x$

And,

  $y_4=y_2-y_1$

Expalnation:

Consider the given functions,

  $y_1=\ln \left(\frac{x}{a}\right),y_2=\ln x$

Find the function $y_3$ by using the above functions.

  $\begin{array}{c}{y}_4=e^{y_3}\\ =e^{y_2-y_1}\\ =e^{\ln a}\\ =a\end{array}$

Now, write the function for the value of $a=2,3,4,5$ .

  $\begin{array}{l}{y}_4=2\left[a=2\right]\\ y_4=3\left[a=3\right]\\ y_4=4\left[a=4\right]\\ y_4=5\left[a=5\right]\end{array}$

Draw the graph of the model by using the above points.

  

As it can be observed in the graph, the functions are the constant function.

Therefore, the function $y_4$ is the constant function.

d.

To determine

To calculate: The value of $y_1$ from the equation $e^{y_2-y_3}=a$ .

Answer to Problem 58E

The result for $y_1$ is $y_1=y_2-\ln a$ .

Explanation of Solution

Given information:

The equation is $e^{y_2-y_1}=a$ .

Formula Used:

The logarithm properties is defined as,

  $\ln e^x=x$

Calculation:

Consider the given equations,

  $e^{y_2-y_1}=a$

Take the log on both sides.

  $\begin{array}{c}\ln e^{y_2-y_1}=\ln a\\ \left(y_2-y_1\right)\ln e=\ln a\\ y_2-y_1=\ln a\\ y_2-\ln a=y_1\end{array}$

Therefore, the obtained result is $y_1=y_2-\ln a$ .