### Instruction: The lengths of the sides of a 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 $n$.### Response: I need to use the fact that the sum of any two sides of a triangle must be greater than the third side, otherwise the triangle would have zero area or be degenerate.
This means that $\log_{10}12 + \log_{10}75 > \log_{10}n$, $\log_{10}12 + \log_{10}n > \log_{ on}75$, and $\log_{10}75 + \log_{10}n > \log_{10}12$.
I can use the properties of logarithms to rewrite these inequalities as $\log_{10}(12\cdot 75) > \log_{10}n$, $\log_{10}(12\cdot n) > \log_{10}75$, and $\log_{10}(75\cdot n) > \log_{10}12$.
Exponentiating both sides with base 10, I get $12\cdot 75 > n$, $12\cdot n > 75$, and $75\cdot n > 12$.
These inequalities can be simplified to $900 > n > 6.25$.
Since $n$ has to be a positive integer, the only possible values for $n$ are 7, 8, 9, ..., 899.
There are 893 such values, so that is my answer.