Solve for $$ x $$ in the following equation: $$ e^{x}-e^{-x}=3 $$

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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:
  $$\left(e^{2x}-1-3e^{x}ight)\left(e^{x}ight)^{-1}=0$$
- step3: Rewrite the expression:
  $$\frac{e^{2x}-1-3e^{x}}{e^{x}}=0$$
- step4: Cross multiply:
  $$e^{2x}-1-3e^{x}=e^{x}	imes 0$$
- 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:
  $$t=\frac{3\pm \sqrt{\left(-3ight)^{2}-4\left(-1ight)}}{2}$$
- 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 $$x:$$
  $$\begin{align}&x=\ln{\left(3+\sqrt{13}ight)}-\ln{\left(2ight)}\&x 
otin \mathbb{R}\end{align}$$
- step13: Find the union:
  $$x=\ln{\left(3+\sqrt{13}ight)}-\ln{\left(2ight)}$$
The solution to the equation $$e^{x}-e^{-x}=3$$ is $$x=\ln{(3+\sqrt{13})}-\ln{(2)}$$.
$$ x = \ln{(3 + \sqrt{13})} - \ln{(2)} $$

Solve for in the following equation:

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