Solve for \( t \) \[ \log _{9}(9 t+4)-\log _{9} t=\log _{9} 10 \] \( t=\square \)

To solve for \( t \) in the equation \[ \log_{9}(9t + 4) - \log_{9} t = \log_{9} 10, \] we can use the property of logarithms that states \( \log_a b - \log_a c = \log_a \frac{b}{c} \). This allows us to rewrite the left-hand side: \[ \log_{9} \left( \frac{9t + 4}{t} \right) = \log_{9} 10. \] Since the logarithms are equal, we can set the arguments equal to each other: \[ \frac{9t + 4}{t} = 10. \] Now, we can multiply both sides by \( t \) (assuming \( t \neq 0 \)) to eliminate the fraction: \[ 9t + 4 = 10t. \] Next, we rearrange the equation to isolate \( t \): \[ 4 = 10t - 9t, \] \[ 4 = t. \] So, the solution is \[ t = 4. \] Thus, the final answer is \[ t = 4. \]