The lengths of the sides of a [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) with positive area are $\log_{10} 12$, $\log_{10} 75$, and $\log_{10} n$, where $n$ is a positive integer. Find the number of possible values for $n$.
By the [Triangle Inequality](https://artofproblemsolving.com/wiki/index.php/Triangle_Inequality) and applying the well-known logarithmic property $\log_{c} a + \log_{c} b = \log_{c} ab$, we have that


$\log_{10} 12 + \log_{10} n > \log_{10} 75$
$\log_{10} 12n > \log_{10} 75$
$12n > 75$
$n > \frac{75}{12} = \frac{25}{4} = 6.25$


Also, 


$\log_{10} 12 + \log_{10} 75 > \log_{10} n$
$\log_{10} 12\cdot75 > \log_{10} n$
$n < 900$


Combining these two inequalities: 
\[6.25 < n < 900\]
Thus $n$ is in the set $(6.25 , 900)$; the number of positive integer $n$ which satisfies this requirement is $\boxed{893}$.