Answer: 893

Small Hint:

The three lengths must satisfy the triangle inequality; two of the three conditions bound $\log_{10} n$ from both sides.

Big Hint:

$\log 75 - \log 12$ $\lt \log n \lt$ $\log 75 + \log 12,$ so $\frac{25}{4} \lt n \lt 900.$

Solution:

The triangle inequality requires $\log n \lt \log 12 + \log 75 = \log 900$ and $\log 12 + \log n \gt \log 75,$ that is $\log n \gt \log 75 - \log 12 = \log \frac{25}{4}.$ The remaining inequality, $\log 75 + \log n \gt \log 12,$ is automatic because $n \ge 1$ and $75 \gt 12.$

So $\frac{25}{4} \lt n \lt 900,$ which for integers means $7 \le n \le 899.$ That gives $899 - 7 + 1 = 893$ possible values of $n.$