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Show that $\log_7 n$ is either an integer or an irrational number where n is a positive number.

I assumed that it is rational and tried to get a contradiction for $\log_7 n = a/b$, where b does not divide a, but how can I show that $7^{a/b}$ is not an integer to achieve a contradiction since n is an integer ? If I can exclude rational numbers from the range of log function then it is either integer or irrational.

Or do you suggest other methods ?

  • 1
    See [this](http://math.stackexchange.com/questions/166441) as well.2012-08-05
  • 0
    As stated your statement is not true. For instance if $n=\sqrt{7}$, then $\log_7\sqrt{7}=\frac{1}{2}$. I think that you need $n$ to be a non-zero natural number.2014-01-02

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