My question is: What is the number of digits of this number? : $$2^{333111160}$$
Asked
Active
Viewed 225 times
0
-
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
2 Answers
5
$$2^{333111160} \sim 10^{(\log_{10} 2) 333111160} \sim 100266459\text{ digits}$$
user_of_math
- 4,320
- 14
- 34
-
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
-
@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
-
-
In at least one expression, you had only three $1$s. I changed it. – Michael Hardy Oct 14 '14 at 18:03
3
$$ 333111160 = \log_2 2^{333111160} = \frac{\log_{10}333111160}{\log_{10}2} $$ $$ \text{So }\log_{10} 333111160 = 333111160\log_{10}2 \approx 100276451.05\ldots $$ This is between $100276451$ and $100276452$, so the number has $100276452$ digits.
But this answer is only as good as the calculator I used.