Given, log₁₂ 27 = a

=> log 27/log 12 = a

=> log 3³ /log(3 * 2² ) = a

=> 3 * log 3 /(log 3 + log 2² ) = a

=> 3 * log 3 /(log 3 + 2*log 2) = a

=> (log 3 + 2*log 2)/3 * log 3 = 1/a

=> log 3/3 * log 3 + 2*log 2 /3 * log 3 = 1/a

=> 1/3 + (2/3)*(log 2/ log 3) = 1/a

=> (2/3)*(log 2/ log 3) = 1/a - 1/3

=> (2/3)*(log 2/ log 3) = (3 - a)/3a

=> log 2/ log 3 = {(3 - a)/3a} * (3/2)

=> log 2/ log 3 = (3 - a)/2a

=> log 3/log 2 = 2a/(3 - a)

=> log 3 = {2a/(3 - a)}*log 2   ...............1

Now, log₆ 16 = log 16 / log 6

=> log₆ 16 = log 2⁴ / log (2 * 3)

=> log₆ 16 = 4*log 2 / (log 2 + log 3)

=> log₆ 16 = 4*log 2 / [log 2 + {2a/(3 - a)}*log 2]        {from equation 1}

=> log₆ 16 = 4*log 2 / [log 2 * {1 + 2a/(3 - a)}]

=> log₆ 16 = 4 / {1 + 2a/(3 - a)}

=> log₆ 16 = 4 / {(3 - a + 2a)/(3 - a)}

=> log₆ 16 = 4 / {(3 + a)/(3 - a)}

=> log₆ 16 = 4*(3 - a)/{(3 + a)