If \( 3^{x^{2}}-1=94 \) find the value of \( x \)

To solve the equation \( 3^{x^{2}} - 1 = 94 \), we first add 1 to both sides to isolate the exponent: \[ 3^{x^{2}} = 95 \] Next, we take the logarithm of both sides, using either common logarithm (base 10) or natural logarithm (base e). We'll go with natural logarithm for this case: \[ \ln(3^{x^{2}}) = \ln(95) \] By the power rule of logarithms, we can simplify the left side: \[ x^{2} \cdot \ln(3) = \ln(95) \] To solve for \( x^{2} \), we divide both sides by \( \ln(3) \): \[ x^{2} = \frac{\ln(95)}{\ln(3)} \] Finally, we take the square root of both sides to find \( x \): \[ x = \pm \sqrt{\frac{\ln(95)}{\ln(3)}} \] Calculating this will yield the values for \( x \).