Starting with the claim 

>! $$ 4^{5^9} > 5^{6^8} $$

We have

>! $$ \iff 5^9\ln4 > 6^8\ln5 $$
>! $$ \iff \frac{5^9}{6^8} > \frac{\ln5}{\ln4} =  1.161 $$
>! $$ \iff 6\times(5/6)^9 > \frac{\ln5}{\ln4} $$
>! $$ \iff \ln(6)+9\ln(5/6) > \ln\frac{\ln5}{\ln4} $$
>! Using $\ln x = - \ln(1/x)$,
>! $$ \iff \frac{\ln6 - \ln\frac{\ln5}{\ln4}}{\ln(6/5)} > 9 $$

Using the hints we have
  
>! $$ \frac{\ln5}{\ln4} = 1.161 $$
>! $$ \ln(4) = 2\ln(2) = 2\times0.6931 $$
>! $$ \ln(6) = \ln(2)+\ln(3) = 0.6931+1.0986 = 1.7917 $$
>! $$ \ln(1.161) = \ln(1+0.161) < 0.161 - 0.161^2/2 + 0.161^3/3 = 0.1494 $$
>! $$ \ln(6) = 1.7917 $$
>! $$ \ln(6/5) = \ln(6)-\ln(5) = 1.7917 - 1.6094 = 0.1823 $$    

we get that the claim

>! $$ \iff (1.7917 - 0.1494)/0.1823 > 9$$
>! $$ \iff 9.0087 > 9$$ 

which is true