Starting with the claim
$$ 4^{5^9} > 5^{6^8} $$
$$ <=> 5^9*ln4 > 6^8*ln5 $$ $$ <=> 5^9 / 6^8 > ln5/ln4 = 1.161 $$ $$ <=> 6*(5/6)^9 > ln5/ln4 $$ $$ <=> ln(6)+9*ln(5/6) > ln[ln(5)/ln(4)] $$ $$ <=> 9 < [ln[ln(5)/ln(4)] -ln6]/ln(5/6) $$
Using the hints we have
$$ ln(5)/ln(4) = 1.161 $$ $$ ln(4) = 2*ln(2) = 2*0.6931 $$ $$ ln(6) = ln(2)+ln(3) = 0.6931+1.0986 = 1,7917 $$ $$ ln(1.161) = ln(1+0.161) < 0.161 - 0.161^2/2 + 0.161^3/3 = 0,1494 $$ $$ ln(6) = 1,7917 $$ $$ ln(5/6) = ln(5)-ln(6) = 1.6094-1,7917 = -0,1823 $$
we get $$ <=> 9 < (0,1494-1,7917)/-0,1823 $$ $$ <=> 9 < 9,0087 $$