We know the rules of logarithms:
log a^b = b * log a
log a * b = log a + log b
and we know that : 10^0.5 = 3,166227766
So:
( log (2) 4 + log (2) 3,16227766) / ( log (2) 20 + log (2) 8 ) =
= ( 2 + log (2) 10^(0.5) ) / ( log (2) 2 + log (2) 10 + 3 ) =
= ( 2 + 0.5 log (2) 10 ) / ( 1 + log (2) 10 + 3 ) =
= ( 2 + 0.5 log (2) 10 / ( 4 + log (2) 10 ) =
= ( 2 + 0.5 log (2) 10 ) / 2 * ( 2 + 0.5 log (2) 10 ) = ( cancelling )
= 1 / 2 = 0.5
