Calculate $2^{1500}$ Is there a way of approximating $2^{1500}$ using pen and paper? I just took $2^{1500}$ as an example but it can be $2^{150000}$ or $12^{95}$. I am just looking for pen and paper method.
By approximate, I mean a solution like $3.5*10^{21}$ or something along those lines.
I was thinking of doing the following: $2^{1500} =x$ then take the $log$ of both sides but I don't know if that makes sense.

It depends how much energy you have.  In the old days a few people calculated things of this size (452 decimal digits).  Repeated squaring can get you $2^{1024}$ in just $10$ multiplications, then you pick the seven you need and multiply them and you are there.  Of course, some of these multiplications are a lot of work.
If you are happy with approximations, going through logs makes it pretty easy.  It helps to have some log tables available.
