I have been studying logarithms from my book. It is a very short chapter (just 5 pages) in the book.
While I was studying it, a question hit my mind: if someone asks me $\log_2(8)$,I'd be able to say 3,
if he asks me $\log_2(32)$, I'd be able to say 5.
But what if he calculates 2^36 on his calculator (which is 68719476736) and asks me $\log_2(68719476736)$; if I don't have a calculator at that time,
would I be able to answer this one?
So my question is to know whether there is any way to get the values of things like $\log_{2}(33554432)$ without using the calculator?
and if there is, what is the method?
Answer
If you know they are asking for the $\log_2$ of a power of $2$ you know the answer is a natural and you just need to find which one. The fact that $2^{10} \approx 10^3$ makes it easy for small numbers like $68719476736$. I would see the leading $5$ digits as not too different from $2^{16}=65536$ and note that there are six more digits, so it must be $2^{16} \cdot (2^{10})^2=2^{36}$
If you don't know the answer is a natural you have to make an approximation. Again $2^{10} \approx 10^3$ is your friend. Your other friend is $\log_{10} (2)\approx 0.30103$. If I want $\log_2 (33554432)$ in my head I would say $33554432\approx 33.5 \cdot 10^6$ so $\log_2 33554432 \approx \log_2(32 \cdot 10^6) \approx 25$. This turns out to be exact as well, but I had assumed you had just mashed the keyboard and we weren't give that the answer was a natural.
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