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

Solve the equation \( e^{x}-e^{-x}=3 \). Solve the equation by following steps: - step0: Solve for \(x\): \(e^{x}-e^{-x}=3\) - step1: Move the expression to the left side: \(e^{x}-e^{-x}-3=0\) - step2: Factor the expression: \(\left(e^{2x}-1-3e^{x}\right)\left(e^{x}\right)^{-1}=0\) - step3: Rewrite the expression: \(\frac{e^{2x}-1-3e^{x}}{e^{x}}=0\) - step4: Cross multiply: \(e^{2x}-1-3e^{x}=e^{x}\times 0\) - step5: Simplify the equation: \(e^{2x}-1-3e^{x}=0\) - step6: Use substitution: \(t^{2}-1-3t=0\) - step7: Rewrite in standard form: \(t^{2}-3t-1=0\) - step8: Solve using the quadratic formula: \(t=\frac{3\pm \sqrt{\left(-3\right)^{2}-4\left(-1\right)}}{2}\) - step9: Simplify the expression: \(t=\frac{3\pm \sqrt{13}}{2}\) - step10: Separate into possible cases: \(\begin{align}&t=\frac{3+\sqrt{13}}{2}\\&t=\frac{3-\sqrt{13}}{2}\end{align}\) - step11: Substitute back: \(\begin{align}&e^{x}=\frac{3+\sqrt{13}}{2}\\&e^{x}=\frac{3-\sqrt{13}}{2}\end{align}\) - step12: Solve the equation for \(x:\) \(\begin{align}&x=\ln{\left(3+\sqrt{13}\right)}-\ln{\left(2\right)}\\&x \notin \mathbb{R}\end{align}\) - step13: Find the union: \(x=\ln{\left(3+\sqrt{13}\right)}-\ln{\left(2\right)}\) The solution to the equation \(e^{x}-e^{-x}=3\) is \(x=\ln{(3+\sqrt{13})}-\ln{(2)}\).