1. You may not use a calculator or computer.

2. You may write "ln(X)" or "log(X)" to indicate the natural logarithm of X.
 Else, please let the reader know "log(X)" means   log of X to the base 10, just to mention another common base.

3. You are allowed to use $ \ ln(1 + x)  \ \approx \ x - \dfrac{x^2}{2} + \dfrac{x^3}{3} \ $ for appropriate small values of $ \ x.$

4. To reduce some arithmetic, you are allowed to use these if they were to come up in calculations:
 
   $ln(2) \ \approx \ 0.6931 $

   $ln(3) \ \approx \ 1.0986 $

   $ln(5) \ \approx \ 1.6094 $

5. If it were to come up, you may use $ \ \dfrac{ln(5)}{ln(4)} \ \approx \ 1.161 \ $ in a calculation.

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Please show the steps without using a calculator or computer to indicate which expression is larger:

$\large4^{5^9} \ \  \ or \ \ \  5^{6^8}$