Log_2aa=x    then, a=(2a)^x  ......(1)

Log_3a2a=y    then,2a=(3a)^y ......(2)

Log_4a 3a=z  then, 3a=(4a)^z ......(3)

So, 

a=(2a)^x  [from (1)]

Or, a=(3a)^xy    [from(2)]

Or, a=(4a)^xyz     [from(3)]

Multiplying both sides by 4a,

4a.a=4a.(4a)^xyz  

Or,(2a)² =(4a)^{xyz + 1} 

Or,(3a)^2y =(4a)^xyz+1 

Or,(4a)^2yz =(4a)^xyz+1 

Or, 2yz = xyz+1 .proved.

Well, the propagation of light is fundamentally due to oscillation of electric field and magnetic field perpendicularly, which allows the light waves to propagate in the direction perpendicular to both the existing fields, i.e. light waves propagation solely depends on these fields.

But, sound waves are non other than the transference of energy due to disturbance, as well the propagation of sound wave has already been found to be adiabatic in nature, and through the adiabatic equation,...

Program to input any number and display number of odd numbers in it;

#include<stdio.h>

#include<stdlib.h>

#include<conio.h>

#include<string.h>

int main() {

int i, number, num1, num2=0, num3, num4 =0, rem, rem1, rem2, rev = 0;

printf("Enter your number ==> ");

scanf("%d", &number);

num1 = number;

while(num1 != 0) {

rem = num1%10;

rem1 = rem%2;

if (rem1 != 0) {

num2 = num2*10 + rem;

}

num1 /= 10;

}

num3 = num2;

while (num2 != 0) {

...

No, if three vectors do not lie in a plane, they cannot give zero resultant.

Explanation:

Let A, B and C be three vectors. If they give zero resultant, then

     A+B+C=0

or, A= -(B+C)

Hence, they will produce zero resultant, if A is equal to negative of vector (B+C). The vector (B+C) lies in the plane of B and C. Hence, A will be equal to negative of (B+C) if A, B and C all lie in a plane.