Question

Prove that log 2 24/ log 96 2 - log 2 192 / log 12 2 =3.

Answer 1

Given: { log ( base 2) 24 / log ( base 96) 2 } - { log ( base 2) 192 / log ( base 12) 2 } = 3

To find: Prove LHS = RHS

Solution:

• Before proceeding, we know the property that log (base b) a = log a / log b.

• Now, lets consider LHS,

= { log ( base 2) 24 / log ( base 96) 2 } - { log ( base 2) 192 / log ( base 12) 2 }

• By using the above property, we get:

 = { log 24 / log 2 / log 2 / log 96 } - { log 192 / log 2 / log 2 / log 12 }

• Now, solving further, we get:

 = { log 24 x  log 96 /( log 2 )²  -  log 192 x  log 12/ (log 2 )²}

 =  { log 24 x  log 96 - log 192 x  log 12 } / (log 2 )²

 =  { log (2³x3) x  log (2^5 x 3) - log (2^6 x 3) x  log (2²x3) } / (log 2 )²

• Now we have the property that

  log (ab) = log a + log b

=log 2³ + log 3 x  log 2^5 + log 3 - (log 2^6 + log 3 x  log 2² + log 3) /(log 2 )²

• Now we have the property that

   log a^b = b log a

=3log 2 + log 3 x  5log 2 + log 3 - (6log 2 + log 3 x  2log 2 + log 3)  / (log 2 )²

= { 15(log 2)² + 3(log 2)(log 3) + 5(log 2)(log 3) + (log 3)² } - { 12(log 2)² + 6(log 2) (log 3) + 2(log 2)(log 3) + (log 3)² } /  (log 2 )²

• After subtracting the terms we get:

 = 15(log 2)² -  12(log 2)² / (log 2 )²

 = 3(log 2 )²/ (log 2 )²

 = 3           ..................RHS

hence proved.

Answer:

        So from above, we have proved that LHS = RHS.