Chapter 4, Problem 1R

Explanation of Solution

Graph of functions using logarithmic scale:

8n:

Let the function be:

$f\left(n\right)=8n$

Take the logarithm on both the sides of the above function as follows:

$logf\left(n\right)=log8n$

$\log f(n)=\log n+\log 8$        (1)

According to the given data, let us take $logf\left(n\right)=Y$ , $logn=\text{ X}$ . Substitute the values in Equation (1):

$Y=X+\log 8$        (2)

Thus, from the above Equation (2), the value of $Y_{interval}=log8$ , $X_{interval}\text{ =}-log8$

The following is the graph representation corresponding to the values $Y_{interval}=log8$, $X_{interval}\text{ =}-log8$ on respective x-axis and y-axis respectively.

4n logn:

Let the function be:

$f\left(n\right)=4n\log n$

Take the logarithm on both the sides of the above function as follows:

$logf\left(n\right)=log\text{ 4}n\log n$

$\log f(n)=\log 4+\log (n\log n)$

$\log f(n)=\log 4+\log n+\log (\log n)$        (3)

According to the given data, let us take $logf\left(n\right)=Y$ , $logn=\text{ X}$ . Substitute the values in Equation (3)

$Y=X+\log (X)+log4$        (4)

The following is the graph representation corresponding for the function $f\left(n\right)=4n\log n$ on respective x-axis and y-axis respectively.

$2n^2$:

Let the function be:

$f\left(n\right)=2n^2$

Take the logarithm on both the sides of the above function as follows:

$logf\left(n\right)\text{ = }log{\text{ 2n}}^2$

$\log f(n)=\log 2+\log n^2$

$\log f(n)=\log 2+2\log n$        (5)

According to the given data, let us take $logf\left(n\right)=Y$ , $logn=\text{ X}$ . Substitute the values in Equation (5)

$Y=2X+log2$        (6)

Thus, from the above Equation (6), the value of $Y_{interval}=log\text{ 2}$ while substituting X =0, $X_{interval}\text{ =}\frac{-log\text{ 2}}{2}$ while substituting Y=0.

The following is the graph representation corresponding to the values $Y_{interval}=log\text{ 2}$, $X_{interval}\text{ =}\frac{-log\text{ 2}}{2}$ on respective x-axis and y-axis respectively.

$n^3$:

Let the function be:

$f\left(n\right)=n^3$

Take the logarithm on both the sides of the above function as follows:

$logf\left(n\right)\text{ = }log{\text{ n}}^3$

$\log f(n)=3\log n$        (7)

According to the given data, let us take $logf\left(n\right)=Y$ , $logn=\text{ X}$ . Substitute the values in Equation (6)

$Y=3X$        (8)

The following is the graph representation corresponding to the value on respective x-axis and y-axis respectively.

$2^n$:

Let the function be:

$f\left(n\right)=2^n$

Take the logarithm on both the sides of the above function as follows:

$logf\left(n\right)\text{ = }log{\text{ 2}}^n$

$\log f(n)=n\log 2$

$\log f(n)=e^{\log n}\log 2$        (7)

According to the given data, let us take $logf\left(n\right)=Y$ , $logn=\text{ X}$ . Substitute the values in Equation (6)

$Y=e^x\log 2$        (8)

The following is the graph representation corresponding to the value on respective x-axis and y-axis respectively.