How do I give a good estimate for $3^{3.8}$ or $3^{308}$ or $3^{3.7}$ etc. quickly and without a calculator? How do I give a good estimate for $3^{3.8}$ or $3^{3.08}$ or $3^{3.7}$ etc. quickly and without a calculator?
Back of the envelope estimate of $3^{3.8}$ to follow:

We know $\log 3$ is slightly bigger than $1$.

Let $f(x) = 3^x$, so that $f'(x) = 3^x \log 3$.  

If $x = 4$ and $dx = -.2$ then $y = 81$ and $dy = 3^4 \log 3 (-.2) = -16.2 (\log 3)$. 

So a not-too-accurate guess using differentials could be $81 - 16.2(\log 3)$. Since $\log 3$ is slightly bigger than $1$ let's just call it $81 - 17 = 64$.

If you go a bit further and use the fact that $y = 3^x$ is convex, we know the initial estimate is going to be too low, so perhaps bump it up a bit to 65.
Do you know how these things are calculated in a calculator, they use the taylor approximation. 

https://en.wikipedia.org/wiki/Taylor_series

So if you could memorize up the first few terms of the series of $a^x$

$a^x = 1 + x log(a) +\frac{1}{2} x^2 log^2(a) + \frac{1}{6}x^3 log^3(a)....$

https://www.wolframalpha.com/input/?i=expand+a%5Ex%5C

and if you know the $log_{10}$ values, this can be used.