<s>[INST] Is there an algorithm to find the number of digits in 2^n, where n is a positive integer? Is there an algorithm to find the number of digits in 2^2030 ?
$2^1=2$
$2^2=4$
$2^3=8$
$2^4=16$
$2^5=32$
$2^6=64$
$2^7=128$
...
$2^{10}=1024$
 [/INST] Just for fun, let's suppose you don't have access to floating-point computations, but you can compute with rational numbers.
We have the series (for $|x|<1$)
$$ \ln \left( \frac{1+x}{1-x}\right) = 2 x + \frac{2}{3} x^3 + \frac{2}{5} x^5 + \frac{2}{7} x^7 + \ldots$$
and in particular, if $0 < x < 1$,
$$  2 x + \frac{2}{3} x^3 + \frac{2}{5} x^5 + \frac{2}{7} x^7 
< \ln\left( \frac{1+x}{1-x}\right) < 2 x + \frac{2}{3} x^3 + \frac{2}{5} x^5 +  
\frac{2 x^7}{7(1 - x^2)}$$
For $x = 1/3$ this gives us
$$ \dfrac{53056}{76545} < \ln(2) < \dfrac{23581}{34020}$$
while for $x = 1/9$ we have
$$ {\frac {37355104}{167403915}} < \ln  \left( 5/4 \right) < {\frac {
3689393}{16533720}}
$$
Then $2030 \log_{10}(2) = \dfrac{2030 \ln 2}{\ln(5/4) + 3 \ln(2)}$ is between
$\dfrac{23263994880}{38070491}$ and $\dfrac{94221399069}{154182208}$, both of which are between $611$ and $612$.
So, $612$ digits.
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