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I like to think of logarithms this way: $\log_a(N)=y$ implies that, the a denotes the power at which you're expanding / shrinking, the N represents the size of you after y units of time has passed.

For example, if I'm expanding two fold by whatever amount I had before with me at t=0 (I assume this typically to be 1), after 1 unit of time has passed, my size is 2, then after 2 units of time has passed, my size is 4 and so on.

The point is, y gives the time you've expanded for if you feed it the other two information. But this infact has an interesting consequence, whatever base I choose (here, 10), if I make note of the final size of the initial 1 I began with as time passes, I literally form the place value system for base 10 (1, 10, 100, 1000....). And so today I learnt that if we wanted to get the number of digits in a number N (represented in base 10), we could just do so by $\lfloor{(\log_{10}(N)+1\rfloor})$. But I really cannot figure out as to why it should be as such.

Does there exist a nice explanation which can be deduced from the way I mentioned I like to think of logarithms as? If we continue this discussion, there also exists this formula to calculate the number of digits in a number when written in the binary form, which is just the change of base from 10 to 2, $\lfloor{(\log_{2}(N)+1\rfloor})$. And I would also like to know about how does logarithms have a relation to the corresponding number systems?

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You have essentially answered your own question with the sequence $$ 1= 10^0, 10=10^1, 100=10^2, 1000=10^3, \ldots . $$ In your excellent exponential growth model for understanding logarithms, each time step multiplies the size by $10$, so adds a $0$ to the decimal representation. After, say, $2.5$ steps your size will be between $10^2 = 100$ and $10^3 = 1000$. Those are precisely the numbers with three digits before the decimal point in base $10$. In general, the number of digits in a positive integer $n$ tells you which pair of powers of $10$ it lies between. The formula you "learned" says just that.

Edit in response to a comment from the OP.

In the base $b$ number system, $1$ followed by a string of $n$ $0$'s is $$ 1 \times b^n + 0 \times b^{n-1} + \cdots +0 \times b + 0 = b^n $$ so the numbers with $n$ digits are the ones between $b^{n-1}$ and $b^n$. These are the numbers whose base $b$ logarithms are between $1$ and $n$. I think that is the closest you can get to a "direct relation between logarithms and number system".

Your model of logarithms using growth over time is just an analogy - a good way to understand the inverse relationship between exponentiation and logarithms. If you plot time on the horizontal axis and size on the vertical axis you see exponential growth. If you use a logarithmic scale on the vertical axis the powers of $10$ are equally spaced and the graph for exponential growth is the red line:

enter image description here

https://en.wikipedia.org/wiki/Semi-log_plot

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  • $\begingroup$ Thank you for your answer and I apologize for the late reply. It makes sense to me. But I wanted to know if there exists a direct relation between logarithms and number system (place values, as in, if I used log_2(N), I would get the base 2 number system). And can you explain more to this: "In any number system, 1 followed by a string of 0's is a power of the base so the argument works for any base." And I would also like to know that how time (in my analogy) relates to the digits or is it just a consequence of my analogy? $\endgroup$
    – Floatoss
    Commented Jan 11, 2022 at 3:04
  • $\begingroup$ I still do not understand what does this mean: "In the base b number system, 1 followed by a string of n 0's is: $1×b^{n}+0×b^{n−1}+⋯+0×b+0=b^{n}$." I am sorry if I sound a bit off here, but I don't understand what does it mean. I would appreciate if you provided an example of it. $\endgroup$
    – Floatoss
    Commented Jan 11, 2022 at 15:25
  • $\begingroup$ That is positional notation. In our decimal system the digit string $xyz$ is $x\times 10^2 + y \times 10^1 + z \times 10^0$. In the number we write as "$100$" you have $x=1$ and $y=z=0$ so you get just plain $10^2$. There is nothing special about $10$ in this discussion - it works just as well in any base $b$. For $b=2$ the string "$10000$" is what we would represent as "$16$". I hope this helps. It's the best I can do. $\endgroup$
    – Ethan Bolker
    Commented Jan 11, 2022 at 15:36
  • $\begingroup$ Oh I get it now, I previously did not catch that you were representing the place values. It makes sense now. Thank you again. $\endgroup$
    – Floatoss
    Commented Jan 12, 2022 at 12:58
  • $\begingroup$ If I may, I want to understand how you generalized this: "so the numbers with n digits are the ones between $b^{n−1}$ and $b^{n}.$ And frankly it was a bit hard for me to understand these as I'm new to number systems, I know I'm not helping here, but I would love to understand more to this stuff. It would be great if you point me to any sources or perhaps just explain more to this yourself! :) $\endgroup$
    – Floatoss
    Commented Jan 12, 2022 at 13:15

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