Is there a direct way to determine how many digits a power of 2 will contain without actually performing the multiplication?
An estimation would help as well if there is no absolute solution.
EDIT: In both decimal and binary bases.
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1In what base??? – The Chaz 2.0 Aug 02 '12 at 11:22
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7Chaz, when no base is specified, what could we possible assume but base 10? – Kevin Carlson Aug 02 '12 at 11:25
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3Q. How do you explain the concept of logarithm to a five year old? A. Roughly speaking, a logarithm is like the the number of digits in a decimal number. Ergo, Américo's answer. :) – Aug 02 '12 at 11:41
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1Regarding the Edit, the number of digits (bits, really) of a power of 2 in binary basis is, well... computable. – Did Aug 02 '12 at 12:31
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As it was asked, in base 2 we will find that $2^n$ has $n$ bits. – Ross Millikan Aug 02 '12 at 13:12
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2@Ross: As indicated in my comment (except one needs $n+1$ bits). – Did Aug 02 '12 at 13:14
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If you solve for $x$ the equation $$2^{n}=10^{x}$$ you get the exponent of $10$ $$x=\frac{n\ln 2}{ \ln 10}\approx 0.30103n\qquad \text{(see comment)}$$
Answer to the edit. In binary base since
$$2^{n}=1\cdot 2^{n}+0\cdot 2^{n-1}+\cdots +0\cdot 2^{2}+0\cdot 2^{1}+0\cdot 2^{0},$$
we have $n+1$ bits
$$\left( 2^{n}\right) _{2}=\underset{n+1\text{ bits}}{\underbrace{1\overset{n\text{ 0's}}{\overbrace{0\ldots 000}}}}.$$
Comment. The number $x$ is never an integer because $2^{n}$ can only terminate in $2,4,6$ or $8$. So, as commented by Random832, the number of digits in decimal base is
$$\left\lfloor 1+\frac{n\ln 2}{\ln 10}\right\rfloor =1+\left\lfloor n\,\log _{10}2\right\rfloor ,$$
which is the sequence A034887 in OEIS (Gost's comment).
Américo Tavares
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2I'd have gone to $\lfloor n\frac{\ln 2}{\ln 10}+1\rfloor$, to get directly to a number of digits. Using floor(x+1) rather than ceil to take into account the case where the base is [a power of] 2, not strictly necessary for base 10. – Random832 Aug 02 '12 at 15:27
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There's an estimate of great use in computer science: $2^{10k} \approx 10^{3k}. $ In particular it's always just a bit bigger, so $2^{10}$ has four digits, $2^{20}$ has seven, and so on. I'm not sure how to find cutoffs for lengths not a multiple of 3.
Kevin Carlson
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I'm not sure if there is a proof for this, but there is a repeating pattern that seems to hold true for as large as I can verify.
As mentioned in another answer, a $2^{10}$ step increases the base 10 number by three digits.
It seems within each $2^{10}$ there is a pattern where each base 10 digit will increment on each of these powers of two: $2^{4} \cdot 2^{3} \cdot 2^{3}$.
One could make a pretty efficient function to compute the number of digits using this pattern. Again, not sure how long this pattern holds true for, but the number is pretty large.
Peter
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1a $2^{10}$ step increases the base 10 number by three digits... Often, but not always. – Did Aug 03 '12 at 10:58
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@did Yea, there is no rigor behind that comment. Haha. Although I'm curious, can you show me a counter-example? – Peter Aug 03 '12 at 15:53
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For every $n$ such that $(10^3/2^{10})\cdot10^k\lt2^n\lt10^k$ for some $k$, the step increases by four digits. And there are infinitely such $n$ by ergodicity. – Did Aug 03 '12 at 16:13
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IntegerPart(log(x)+1) where x is some equation will give the number of digits on the left hand side of the decimal. It's what I use anyway :)
Chad Harrison
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