TITLE: approximation for value of $2^x$ without using calculator
QUESTION [5 upvotes]: How to find an approximation for value of $2^x$ without using calculator?
For example, $2^{4.3}$.

REPLY [3 votes]: I suppose you are asking how to do it without a calculator or logarithm tables with a high precision. And how to do it fast. Of course, there is a way. Nowadays we calculate even indefinite integrals with computers. 
Well, here is what I suggest in this particular case:
$$2^{4.3}=2^3\cdot2^{1.3}=8\cdot2^{1.3}$$
Now it is very easy to get a very precise answer fast by using the Taylor's formula for fraction powers of the binomial (1+x):
$$(1+x)^{t}=1+\frac{t}{1!}\,x+\frac{t\,(t-1)}{2!}\,x^2+...+\frac{t\,(t-1)\cdot...\cdot t\,(t-n+1)}{n!}\,x^n+...$$
So, we get this:
$$2^{1.3}=(1+1)^{1.3}=1+\frac{1.3}{1!}\cdot1+\frac{1.3\cdot(1.3-1)}{2!}\cdot1^2+\frac{1.3\cdot(1.3-1)\,(1.3-2)}{3!}\cdot1^3+...$$
or
$$1+\frac{1.3}{1!}+\frac{1.3\cdot(1.3-1)}{2!}+\frac{1.3\cdot(1.3-1)\,(1.3-2)}{3!}+\frac{1.3\cdot(1.3-1)\,(1.3-2)\,(1.3-3)}{4!}...$$
For simplicity of my crude calculations I'm multiplying everything by $1000$:
$$1000+1300+\frac{130\cdot(13-10)}{2!}+\frac{13\cdot(13-10)\,(13-20)}{3!}+\frac{1.3\cdot(13-10)\,(13-20)\,(13-30)}{4!}...$$
Now we get this:
$2300+195-13\cdot49/6+1.3\cdot49\cdot27/24≈2495-650/6+1.3\cdot2\cdot27≈2500-110+70=2460$
Now $\;8\cdot2460/1000=19680/1000=19.68\quad$ The actual number is 19.698...
My calculation here is too crude but you get the idea. I don't even bothered with precise arithmetic here.
As the factorial in the denominator grows larger you get your precision. 
Note how the terms alternate (you either add a term or subtract it). I chose the number $1.3$ on a hunch to sorta "make" a few terms smallish.
I think it is important to understand the ideas, rather than memorize things like this: 
$$\sqrt[3]{1+x}=1+\frac{1}{3}\,x-\frac{2}{2!\cdot3^2}\,x^2+\frac{2\cdot5}{3!\cdot3^3}\, x^3-\frac{2\cdot5\cdot8}{4!\cdot3^4}\, x^4+...$$
PS: You can also calculate sines, cosines, logarithms etc. in a similar fashion that is "on paper" without logarithm tables or calculators.