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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?

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  • $\begingroup$ Either you have a $\ln$ function available and can directly calculate $\dfrac{\ln(x)}{\ln(2)}$ either you loop dividing $x$ by $2$ if even or dividing $x+1$ by $2$ until you reach $1$. $\endgroup$ – zwim Oct 3 '17 at 4:31
  • $\begingroup$ @alonso thanks for the edit alonso,i guess i'm a rookie at writing these things. $\endgroup$ – Durgeshwar Ojha Oct 3 '17 at 4:57
  • $\begingroup$ You are very welcome @DurgeshwarOjha . You'll get better at this sooner than you think. Welcome to math.se! $\endgroup$ – Alonso Delfín Oct 3 '17 at 5:01
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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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  • $\begingroup$ thanks man,it is a really good way of telling them the answer,this 2^10 approx. equal to 10^3 thing,but how did my other friend log of 2 to the base 10 approx equal to 0.30103 help me? $\endgroup$ – Durgeshwar Ojha Oct 3 '17 at 4:52
  • $\begingroup$ It didn't here. It could if you wanted a nonnatural value. It gives you $\log_2 (10) \approx 3.33$, so $\log_2(100) \approx 6.66$. You can use this for values between powers of $2$. $\endgroup$ – Ross Millikan Oct 3 '17 at 5:04

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