My question is: What is the number of digits of this number? : $$2^{333111160}$$
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1Related : http://math.stackexchange.com/questions/177973/is-there-a-way-to-determine-how-many-digits-a-power-of-2-will-contain – lab bhattacharjee Oct 14 '14 at 17:47
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$$2^{333111160} \sim 10^{(\log_{10} 2) 333111160} \sim 100266459\text{ digits}$$
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Why did you set part of your answer in MathJax but not all of it? (I changed it.) – Michael Hardy Oct 14 '14 at 17:53
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@MichaelHardy Thanks. I noticed that with a more accurate value of $\log_{10}2$, you get 100276451 digits and I was editing that in. – user_of_math Oct 14 '14 at 17:56
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In at least one expression, you had only three $1$s. I changed it. – Michael Hardy Oct 14 '14 at 18:03
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$$ 333111160 = \log_2 2^{333111160} = \frac{\log_{10}333111160}{\log_{10}2} $$ $$ \text{So }\log_{10} 333111160 = 333111160\log_{10}2 \approx 100276451.05\ldots $$ This is between $100276451$ and $100276452$, so the number has $100276452$ digits.
But this answer is only as good as the calculator I used.