Solve for \( x \) in the following equation: \( e^{x}-e^{-x}=3 \)

To solve the equation \( e^{x} - e^{-x} = 3 \), we can start by rewriting it in terms of the hyperbolic sine function. Recognizing that \( e^{x} - e^{-x} = 2\sinh(x) \), we can rewrite the equation as: \[ 2\sinh(x) = 3 \] Dividing both sides by 2 gives us: \[ \sinh(x) = \frac{3}{2} \] Now we can solve for \( x \) using the inverse hyperbolic sine function: \[ x = \sinh^{-1}\left(\frac{3}{2}\right) \] This can also be expressed in terms of logarithms: \[ x = \ln\left(\frac{3}{2} + \sqrt{\left(\frac{3}{2}\right)^2 + 1}\right) \] Thus, the solution for \( x \) is: \[ x = \ln\left(\frac{3}{2} + \sqrt{\frac{9}{4} + 1}\right) = \ln\left(\frac{3}{2} + \sqrt{\frac{13}{4}}\right) = \ln\left(\frac{3 + \sqrt{13}}{2}\right) \] So, the final answer is: \[ x = \ln\left(\frac{3 + \sqrt{13}}{2}\right) \] And that's the value of \( x \)!