Convert scientific notation decimal number to binary I'm given the number:
$$8.881784197001252 \cdot 10^{-16}$$
I know it is $2^{-50}$ but suppose I didn't know. I need to convert it to binary.

One way is to apply the school taught algorithm on the fractional part. That'd be:

$.881784197001252\cdot2=1.763568394 \to 1$

$.763568394 \cdot 2=1.527136788 \to 1$

$.527136788 \cdot 2=1.054273576 \to 1$

$.054273576 \cdot 2= 0.108547152 \to 0$

$.108547152 \cdot 2=0.217094304 \to 0$

$...$

It may go on forever so I'd need to determine when to stop.

I then thought I could write the number as:
$$8881784197001252 \cdot 10^{-31}$$
Now the number is an integer and I can precisely convert it. It's:
$$11111100011011110111110001000000010001011000000100100$$
The problem is that I don't know how to convert the base and its exponent from base 10 to base 2. I can approximate it after some attempts. It should be between $2^{-100}$ and $2^{-99}$.

So what I'm asking is: what is the correct approach to convert that number? And, if different from the former, what method does the computer use?
There is more than one correct approach to convert the number.  The former is a correct algorithm for the problem.  This kind of base conversion would usually be implemented in a library for the programming language you are using.  There might be many such implementations out there, and I don't know what all of them are doing.

I would not recommend the latter approach, as I expect computing $2^\alpha$ where $\alpha=31 \log_2 10$ will be painful.

See also The math behind converting from any base to any base without going through base 10? and https://math.stackexchange.com/q/48968/14578 and https://en.wikipedia.org/wiki/Positional_notation#Base_conversion and https://en.wikipedia.org/wiki/Binary_number#Conversion_to_and_from_other_numeral_systems.