1) That old man is Aksionov.

2) The speaker is asking about the truth of digging an escape hole .

3) The speaker means the prison's wall.

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.

If a vector A is multiplied by a real number (say n), the vector of same nature is obtained but its magnitude is n times that of A.

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,...