How to compute Euler constant $(e^x)$ to its any power.

Question

How to compute $e^x$ ($2.71218...$) to its any power with any shortcut or a method. I want to know a method to calculate in big powers like $e^{50}$ not small powers, For eg-$0.02$ (using Taylor series or Feymenn method.) If you want to give any alternative method prescribed above for finding small powers, You could give.

Answer 1

Since $$ \cdots \;{4 \over {10}} < {{21} \over {50}} < \log _{10} e = 0.43429 \cdots < {{22} \over {50}} < {4 \over 9} < {5 \over {10}}\; \cdots $$ and you can find many other better bounds, depending on the precision that you need.

Then for instance you can get $$ 10^{\,21} < e^{\,50} < 10^{\,22} $$ or better $$ e^{\,50} \approx 10^{\,21} \cdot 10^{\,{{14} \over {1000}}50} = 10^{\,21} \cdot 10^{\,{7 \over {10}}} = 10^{\,22} \cdot 10^{\, - {3 \over {10}}} \approx 10^{\,22} {1 \over {\root 3 \of {10} }} \approx {1 \over 2}10^{\,22} $$