While studying logarithms with my son, we came across the theorem stating that
$$\log_{10}n$$
is irrational when n is not a power of 10
He noted that if n is not a power of 10, then the logarithm, by definition, should not exist, since the logarithm is that number to which we raise the base to obtain n (definition of logarithm).
To express this more clearly, writing $$ x= \log_{10}n$$
means that x is that number such that $$10^x =n$$ Saying that n is not a power of 10 seems to me like saying that x does not exist, which contradicts the theorem that states that x is irrational. If it does not exist it cannot be irrational
We have though of two possible solutions, neither of which convinces us:
1) For power of 10 we intend only 10 raised to a rational exponent. In this case the statement is circular because it would state that if n is not a rational power of 10 then the logarithm is irrational. This is the same as saying that if the logarithm is irrational then it is irrational. Therefore it does not convince us
2)
The theorem is more correctly stated as
$$\log_{10}n$$
is irrational if n cannot be expressed as a power of 10. Meaning it does exist, but cannot be expressed as a power of 10. This is also unconvincing since if the theorem is proven, it shows that n can indeed be expressed as a power of 10, albeit with an irrational exponent.
Conclusion
I hove found proofs of this theorem. My guess is that expert mathematicians have thought hard about it. As I am a chemist, probably I am missing something. I would be glad for some mathematical support.