TITLE: Solve without a calculator: What is the possible value of 2*((1+1/100)^100)?
QUESTION [5 upvotes]: What is the possible value of $2·((1+\tfrac{1}{100})^{100})$?
Google will give $2·((1+\tfrac{1}{100})^{100}) = 5.40962765884$.
How can I find the possible value without Google or a calculator?
How can I solve an equation like $(x^{100})$  or  $(x^{20000})$?
Or like $(1.01^{100})$?

REPLY [3 votes]: I would go as follows:
Let $y=2(1+\frac{1}{100})^{100}$ so $\mathrm{ln}(y)=\mathrm{ln}(2)+100\mathrm{ln}(1.01)$.
A first-order Taylor expansion around $\mathrm{ln}(1.01)$ is quite accurate and equal to $\mathrm{ln}(1)+1*(1.01-1)=0.01$. Therefore, $\mathrm{ln}(y)=\mathrm{ln}(2)+100*0.01=\mathrm{ln}(2)+1=\mathrm{ln}(2)+\mathrm{ln}(e)=\mathrm{ln}(2e)$.
Finally, undo the transformation in both sides so $y=2e$.