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.

The Wheatstone bridge is not suitable for measuring very low resistance because it is based on a ratio of two resistances, and the resolution of the bridge decreases as the ratio approaches 1. This means that the Wheatstone bridge is not accurate enough to measure very small changes in resistance.

Yes, a physical quantity can have magnitude and direction but still be a scalar if it doesn't obey the vector addition. An example is Electric Current which has magnitude and a fixed direction, but it does not follow vector laws of addition.

Bohr's theory of the atom was proposed in the early 20th century and was based on the idea that electrons move in fixed orbits around the nucleus. According to this theory, the position and momentum of an electron in an atom can be determined with absolute precision at any given moment, which seems to contradict Heisenberg's uncertainty principle.

Heisenberg's uncertainty principle states that it is impossible to determine the exact position and momentum of a particle simultaneously. This...

If vectors are arranged as trigonal planar and have equal magnitude, it is an ideal case of zero resultant.

The two vectors (say A and B) of different magnitudes cannot be combined to give zero resultant since minimum value of combination is ІA-BІ which is not zero if AB.

The three vectors A, B and C of different magnitudes can be zero such that they form a closed triangle, then,

      A+B+C=0

or, C=-(A+B)

Hence, the sum of three vectors may be zero if vector sum of any two vectors is equal and opposite to the third vector.

Note: The vectors can give this result only if...