Interrelationship between power of 2 and integer length

[Nile3]

Hello, I was wondering if someone knew the equation which describe the exact or statistical relation between the length(# of digits) of a power of 2 based on it's power.

I plotted 200 of the powers in mathematica and I get a fairly straightforward staircase plot. I'm just wondering what's the rule here.

 

[R136a1]

The length of an integer $x$ in basis 10 is given by $\lfloor\log_{10}(x)\rfloor+1$, where the strange brackets denote the largest integer smaller or equal than $x$. So if $x=2^n$, then the length is

$$\lfloor \log_{10}(2^n)\rfloor + 1 \sim n\log_{10}(2) + 1$$

 

[JanEnClaesen]

If the number of digits of a number increases linearly, the number increases exponentially; if the exponent of a number increases linearly, the number increases exponentially. Basically you've plotted the implicit function 10^y = 2^x.

y/x = b is a constant because the equation 2 = 10^b has one root. To estimate b consider 2^10 = 1024 and 10^3 = 1000.

 

[Nile3]

n * (memorize 0.3 as log[10,2]) +1, excellent, thanks!

 

[CellCoree]

I can provide some insight into the interrelationship between the power of 2 and the integer length. The equation that describes this relationship is as follows:

Integer length = log(base 10) (power of 2)

In other words, the integer length is equal to the logarithm of the power of 2, using a base of 10. This means that as the power of 2 increases, the integer length also increases, but at a slower rate. This is why you observed a staircase plot in your Mathematica graph.

This relationship is known as a logarithmic relationship, where one variable (in this case, the power of 2) increases exponentially while the other variable (integer length) increases at a slower rate. It is a common relationship in many scientific fields, and is often used to describe the growth of phenomena.

I hope this helps to answer your question and provide some context for the relationship between power of 2 and integer length.