rational numbers

[Imum Coeli]

Hi not sure if this is the right place for this but I have a question that I do not know how to start.

Q: For which positive rational numbers r is log_10 (r) also rational?

I know the definition but that's as far as I can get.

 

[stapel]

I know the definition but that's as far as I can get.

You "know the definition" of what? Logarithms? (Apologies for my confusion.)

Hi not sure if this is the right place for this but I have a question that I do not know how to start.

Q: For which positive rational numbers r is log_10 (r) also rational?

Perhaps it might help to rearrange the statement using The Relationship: You are asked about log_10(r) = q, where q is rational. This is the same thing as:

. . . . .\(\displaystyle 10^q\, =\, r\)

...where r and q are rational. Can you "see" what values might work?

 

[Imum Coeli]

Sorry, that wasn't very clear. I meant definitions of both logarithms and rationals.

Clearly it is true if q is a natural number, but is there a way to tell if I'm missing anything? I think I should use prime factorisations somehow.

 

[HallsofIvy]

Your original question was: For what rational numbers, r, is the logarithm (base 10) of r rational? If q= log(r) then r= 10^q power. If q is rational then q= a/b for integers a and b so that r= 10^(a/b). That will be true as long as r is a power of some root of 10. Since any root of 10 is irrational, the only rational r that have a rational logarithm are the integer powers of 10 which give integer logarithms.

 

[Imum Coeli]

Such a simple argument! Very nice.
Thank you so much.