Chapter 4, Problem 1R

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:

log f(n) = log 8n

logf(n) = log n+ log 8        (1)

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

Y=X+log8        (2)

Thus, from the above Equation (2), the value of Yinterval= log 8 , Xinterval =−log 8

The following is the graph representation corresponding to the values Yinterval= log 8, Xinterval =−log 8 on respective x-axis and y-axis respectively.

4n logn:

Let the function be:

f(n)=4nlogn

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

log f(n) = log 4n log n

logf(n) = log 4+ log (nlog n)

logf(n) = log 4+ log n + log(log n)        (3)

According to the given data, let us take log f(n)=Y , log n= 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.

2n2:

Let the function be:

f(n)=2n2

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

log f(n) = log 2n2

logf(n) = log 2+ log n2

logf(n) = log 2+  2log n         (5)

According to the given data, let us take log f(n)=Y , log n= 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.

n3:

Let the function be:

f(n)=n3

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

log f(n) = log n3

logf(n) = 3log n        (7)

According to the given data, let us take log f(n)=Y , log n= 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.

2n:

Let the function be:

f(n)= 2n

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

log f(n) = log 2n

logf(n) = nlog 2

logf(n) = elog nlog 2        (7)

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

Y= ex  log2        (8)

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