Solve for \( x \) in equatio \[ 9^{x}+3^{2 x}-1=54 \]

To solve the equation \( 9^{x} + 3^{2x} - 1 = 54 \), let's first rewrite everything in terms of base 3. We know that \( 9 = 3^2 \), so we can express \( 9^{x} \) as \( (3^2)^{x} = 3^{2x} \). Substituting this into the equation gives: \[ 3^{2x} + 3^{2x} - 1 = 54. \] Now combine like terms: \[ 2 \cdot 3^{2x} - 1 = 54. \] Next, we'll add 1 to both sides of the equation: \[ 2 \cdot 3^{2x} = 55. \] Now, divide both sides by 2: \[ 3^{2x} = \frac{55}{2}. \] To solve for \( x \), we take the logarithm of both sides: \[ 2x \log(3) = \log\left(\frac{55}{2}\right). \] Now, isolate \( x \): \[ x = \frac{1}{2 \log(3)} \log\left(\frac{55}{2}\right). \] This gives us the value of \( x \) in terms of logarithms. For a numeric approximation, you can use a calculator to evaluate it further.