Graph of functions using logarithmic scale:

8n:

Let the function be:

f(n)=8n

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

logf(n)=log8n

logf(n)=logn+log8 (1)

According to the given data, let us take logf(n)=Y , logn= X . Substitute the values in Equation (1):

Y=X+log8 (2)

Thus, from the above Equation (2), the value of Yinterval=log8 , Xinterval =−log8

The following is the graph representation corresponding to the values Yinterval=log8, Xinterval =−log8 on respective x-axis and y-axis respectively.

Data Structures and Algorithms in Java, Chapter 4, Problem 1R , additional homework tip 1

4n logn:

Let the function be:

f(n)=4nlogn

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

logf(n)=log 4nlogn

logf(n)=log4+log(nlogn)

logf(n)=log4+logn+log(logn) (3)

According to the given data, let us take logf(n)=Y , logn= X . Substitute the values in Equation (3)

Y=X+log(X)+log4 (4)

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

Data Structures and Algorithms in Java, Chapter 4, Problem 1R , additional homework tip 2

2n2:

Let the function be:

f(n)=2n2

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

logf(n) = log 2n2

logf(n)=log2+logn2

logf(n)=log2+2logn (5)

According to the given data, let us take logf(n)=Y , logn= X . Substitute the values in Equation (5)

Y=2X+log2 (6)

Thus, from the above Equation (6), the value of Yinterval=log 2 while substituting X =0, Xinterval =−log 22 while substituting Y=0.

The following is the graph representation corresponding to the values Yinterval=log 2, Xinterval =−log 22 on respective x-axis and y-axis respectively.

Data Structures and Algorithms in Java, Chapter 4, Problem 1R , additional homework tip 3

n3:

Let the function be:

f(n)=n3

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

logf(n) = log n3

logf(n)=3logn (7)

According to the given data, let us take logf(n)=Y , logn= 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.

Data Structures and Algorithms in Java, Chapter 4, Problem 1R , additional homework tip 4

2n:

Let the function be:

f(n)=2n

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

logf(n) = log 2n

logf(n)=nlog2

logf(n)=elognlog2 (7)

According to the given data, let us take logf(n)=Y , logn= X . Substitute the values in Equation (6)

Y=exlog2 (8)

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

Data Structures and Algorithms in Java, Chapter 4, Problem 1R , additional homework tip 5