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:

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