Solve the logarithmic equation \( \log_{5}(t+16) - \log_{5}(t+7) = \log_{5} t \).
Combine the logarithmic expressions.
Using the property of logarithms, we rewrite the left side as \( \log_{5}\left(\frac{t+16}{t+7}\right) = \log_{5} t \).
Set the arguments equal to each other.
This gives us the equation \( \frac{t+16}{t+7} = t \).
Cross-multiply to eliminate the fraction.
We obtain \( t + 16 = t(t + 7) \), which simplifies to \( t^2 + 7t - t - 16 = 0 \) or \( t^2 + 6t - 16 = 0 \).
Solve the quadratic equation.
Factoring gives us \( (t - 2)(t + 8) = 0 \), leading to solutions \( t = 2 \) and \( t = -8 \).
Check the validity of the solutions.
Only \( t = 2 \) is valid since logarithms are defined for positive arguments.
The solution is \( \boxed{2} \).
The solution is \( \boxed{2} \).