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Not a math guy, just asking out of curiosity.

I recently came to understand that a logarithm's integer and fraction parts are respectively called the characteristic and the mantissa (if that's wrong then feel free to smash me to pieces).

Right now I need to do some work with base ten log results and I need to do something depending on if the resulting logarithm has a mantissa of zero:

log10(0.1) = -1
log10(0.2) = -0.698...
...
log10(1) = 0

In the expressions above, is there a specific terminology to differentiate results like -1 and 0 from other logarithms?

Crono
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  • If the number you are taking a log of is an integer power of the base, (in the case of $10$, any number of the form $10^{k}$, i.e $0.1 = 10^{-1}$, $1 = 10^0$, will return the integer $k$ when you take the base 10 $\log$. That is to say $\log_{10}(10^k) = k$. Thus, when $k$ is an integera, your mantissa will be $0$ (i.e you will have no fractional part). – rubikscube09 Dec 15 '18 at 02:19
  • @rubikscube09 And is there a name for those? Or they're not considered any more special than other common logarithms? – Crono Dec 15 '18 at 03:00
  • Just powers of 10? I'm not sure how much more specific I can be. I guess to answer your question, no not really. Unless I am misunderstanding what you may be trying to ask. – rubikscube09 Dec 15 '18 at 04:04
  • Are you asking for a term when output of the logarithm is an integer? Those just represent when the input is a perfect power of the base. And 0, 1,-1 represent, of course, 1, the base, and the reciprical of the base. – fleablood Dec 16 '18 at 16:41
  • @fleablood So is "perfect power" the correct terminology? – Crono Dec 17 '18 at 13:04
  • @rubikscube09 I don't think you are misunderstanding. This is about terminology. Like 2 can be called a number, it also can be called an integer, an even number, a prime number and a square root when it is the result of √4. There might simply not be any specific designation for powers of ten that are "perfect". Hence, that would be the answer to my question. – Crono Dec 17 '18 at 13:17
  • " So is "perfect power" the correct terminology?" I don't have any freaking idea. Why would anyone need terminology for this? – fleablood Dec 17 '18 at 18:42
  • @fleablood I don't mean to disgress but why is there terminology for anything? :-) A logarithm that also happens to be an integer is also the result of an integer power of the smallest possible figure that can be expressed by a finite number of digits (10 for 2, 100 for 3, etc...). Given how large the math glossary is, it didn't seem unreasonable to believe there would also be a way to designate a logarithm that also is a shift to an additionnal digit... – Crono Dec 17 '18 at 20:32

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