How about, "because I'm going to need six 10-digits to get up to a million, and something more than a 2-digit and less than a 3-digit to get from there to a number between 2 and 3 million."
I'm not sure that would be the right way to say it, but I still feel like the current text is problematic, because:
1) Whether you say last digit, or seventh digit, in either case I'm reading right-to-left and my first thought is that you're talking about the ones place.
2) Even if you said something like left-most digit, that wouldn't be right, because it's not that 2 is between 2 and 3, it's that the value of the whole number is greater than 2*10^6 and less than 3*10^6.
I think you're referring to a digit in an abstract sense that doesn't directly map to the digits we write down, so you may have to go out of your way to avoid confusing nth digit with a particular one of the numerals that are written above.
I might write this as, "whereas, when multiplying 1 by 10 to get to x, you might have to multiply by 10 a fractional number of times (if x is not a power of 10), so the log base 10 of x can include a fractional part while the number of digits in the base 10 representation of x is always a whole number."
Rationale: in the previous sentence you're comparing the number of digits needed to write x to the number of times to multiply 1 by 10. So when the next sentence starts with, "the only difference is..." I'm expecting it to be comparing numbers of digits and numbers of times to multiply. I can figure out that you've switched to talking about "computing logs" because logs count the number of times to multiply by 10, but it feels like one extra step of mental effort.
(This is a less confident suggestion than the amount of text I've used suggests.)
You use an example of "99" then switch to "97".
Is this paragraph needed? I find myself wanting to skip past it.
How about, "because I'm going to need six 10-digits to get up to a million, and something more than a 2-digit and less than a 3-digit to get from there to a number between 2 and 3 million."
I'm not sure that would be the right way to say it, but I still feel like the current text is problematic, because:
1) Whether you say last digit, or seventh digit, in either case I'm reading right-to-left and my first thought is that you're talking about the ones place.
2) Even if you said something like left-most digit, that wouldn't be right, because it's not that 2 is between 2 and 3, it's that the value of the whole number is greater than 2*10^6 and less than 3*10^6.
I think you're referring to a digit in an abstract sense that doesn't directly map to the digits we write down, so you may have to go out of your way to avoid confusing nth digit with a particular one of the numerals that are written above.
I might write this as, "whereas, when multiplying 1 by 10 to get to x, you might have to multiply by 10 a fractional number of times (if x is not a power of 10), so the log base 10 of x can include a fractional part while the number of digits in the base 10 representation of x is always a whole number."
Rationale: in the previous sentence you're comparing the number of digits needed to write x to the number of times to multiply 1 by 10. So when the next sentence starts with, "the only difference is..." I'm expecting it to be comparing numbers of digits and numbers of times to multiply. I can figure out that you've switched to talking about "computing logs" because logs count the number of times to multiply by 10, but it feels like one extra step of mental effort.
(This is a less confident suggestion than the amount of text I've used suggests.)
This is slightly confusing, because it's the first digit that's a 2.
I think you may need to spell out this 10 times as many numbers part. This is a large unexplained step in explaining why the log is the length.
What's n exactly?