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$\frac {1}{10} > \frac {1}{100}$

$ln \frac {1}{10} > ln \frac {1}{100}$

$ln \frac {1}{10} > ln (\frac {1}{10})^2$

$ln \frac {1}{10} > 2 \cdot ln (\frac {1}{10})$

Cancelling $ln(\frac {1}{10})$ , we obtain $1 > 2$

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logarithm of any number less than 1 is negative!