Starting with the claim
$$ 4^{5^9} > 5^{6^8} $$
$$ \iff 5^9\ln4 > 6^8\ln5 $$ $$ \iff 5^9 / 6^8 > \ln5/\ln4 = 1.161 $$ $$ \iff 6*(5/6)^9 > \ln5/\ln4 $$ $$ \iff \ln(6)+9\ln(5/6) > ln[ln(5)/ln(4)] $$ Dividing by $\ln(5/6)<0$, $$ \iff 9 < [\ln[\ln(5)/\ln(4)] -\ln6]/\ln(5/6) $$
Using the hints we have
$$ \ln(5)/\ln(4) = 1.161 $$ $$ \ln(4) = 2\ln(2) = 2\times0.6931 $$ $$ \ln(6) = \ln(2)+\ln(3) = 0.6931+1.0986 = 1.7917 $$ $$ \ln(1.161) = \ln(1+0.161) < 0.161 - 0.161^2/2 + 0.161^3/3 = 0.1494 $$ $$ \ln(6) = 1.7917 $$ $$ \ln(5/6) = \ln(5)-\ln(6) = 1.6094-1.7917 = -0.1823 $$
we get $$ \iff 9 < (0.1494-1.7917)/-0.1823 $$ $$ \iff 9 < 9.0087 $$
