Common logarithm of an integer

Question

I am trying to figure out how to figure out if a common logarithm is an integer or not. For example, the common logarithms 1, 2, 3, are the common logarithms of 10, 100, 1000 respectively. The integers from 10 to 100 are between 1 and 2. So, can I use extrapolation to determine if a common logarithm is a logarithm if an integer or not?

Answer 1

log10^x = x

log10 = 1 as 10^1 = 10

log100 = 2 as 10^2 = 100

1<logN<2 if 10<N<100,

LogM is not an integer if 10^p < M < 10^(p+1)