Question

1) Prove that if loga x = y, then lo8b x = (loga x)(logb a). Plot loga n for 1 S ns 100 when a 2, e and 10.(10-points) 2) Prove that (b) Ση-112-n(n+ 1 )(2n+ 1 )/6 k+1) (1-x) if X Hint: Induction - Discrete Math (15-points) 3) Consider two balanced dices with each of the six faces marked 1 to 6. In one single throw of these two dice, list all the possible outcomes. Compute the average value of the outcomes. (10-points) 4) Compute the size of the problem that can be solved on a machine that takes 10-12 sec/op in one day when T(n) = 4 n, 10 n, 2n . (10-points)

Answer 1

As per the HomeworkLib policy on multiple questions, answer to first question is being given here:

(1)

Given:
log_ax=y Rightarrow x=a^y

To prove
log_bx=(log_ax)(log_ba) ;....(1)

let:
LHS = log_bx
and
RHS = (log_ax)(log_ba)

First, let us take LHS:

Using the given relation (1)

Now, let us take RHS:

Using the given relation (1)

From (2) and (3) we conclude that:
LHS=RHS and hence:

Now coming to the plot (This has been done using MATLAB, the script is also being given at the end for your reference)

Individual Plots for a=2,e and10

log,n,V n-1..100 0 10 20 40 60 70 80 90 100 log n,V n-1..100 10 20 70 80 100 log on,V n-1..100 10 1.5 0.5 10 20 60 70 80 100

Combined Plots for a=2,e and10

og2n logn l0910 0 10 20 30 40 50 60 70 80 90 100 n-1,2,3...100

MATLAB script for your reference:

%=================================

clear all;
clc;

% Given data
n = 1:100;
a2 = log2(n);
ae = log(n);
a10 = log10(n);

% Individual plots
figure
subplot(3,1,1)
plot(n,a2,'-+');
title('log_2{n}, orall n=1..100');
xlabel('n');
ylabel('log_2{n}');
subplot(3,1,2)
plot(n,ae,'-+');
title('log_e{n}, orall n=1..100');
xlabel('n');
ylabel('log_e{n}');
subplot(3,1,3)
plot(n,a10,'-+');
title('log_{10}{n}, orall n=1..100');
xlabel('n');
ylabel('log_{10}{n}');

% Combined plots
figure
plot(n,a2,'-+');
hold on
plot(n,ae,'-+');
hold on
plot(n,a10,'-+');
legend('log_2{n}','log_e{n}','log_{10}{n}');
xlabel('n=1,2,3...100');
ylabel('log_a{n}, orall a={2,e,10}');

%=================================