Is the difference of the natural logarithms of two integers always irrational or 0?

If I have two integers $a,b > 1$. Is

$\ln(a) - \ln(b)$

always either irrational or $0$. I know both $\ln(a)$ and $\ln(b)$ are irrational.

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If $\log(a)-\log(b)$ is rational, then $\log(a)-\log(b)=p/q$ for some integers $p$ and $q$, hence $a/b=\mathrm e^{p/q}$. If $p\ne0$, then $\mathrm e=(a/b)^{q/p}$ is algebraic, which is absurd. Hence $p=0$, and $a=b$.