12. $$ \log _{2}(x+2)+\log _{2}(x-5)=3 $$

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Solve the equation $$ \log_{2}(x+2)+\log_{2}(x-5)=3 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\log_{2}{\left(x+2ight)}+\log_{2}{\left(x-5ight)}=3$$
- step1: Find the domain:
  $$\log_{2}{\left(x+2ight)}+\log_{2}{\left(x-5ight)}=3,x>5$$
- step2: Transform the expression:
  $$\log_{2}{\left(\left(x+2ight)\left(x-5ight)ight)}=3$$
- step3: Convert the logarithm into exponential form:
  $$\left(x+2ight)\left(x-5ight)=2^{3}$$
- step4: Evaluate the power:
  $$\left(x+2ight)\left(x-5ight)=8$$
- step5: Expand the expression:
  $$x^{2}-3x-10=8$$
- step6: Move the expression to the left side:
  $$x^{2}-3x-10-8=0$$
- step7: Subtract the numbers:
  $$x^{2}-3x-18=0$$
- step8: Factor the expression:
  $$\left(x-6ight)\left(x+3ight)=0$$
- step9: Separate into possible cases:
  $$\begin{align}&x-6=0\&x+3=0\end{align}$$
- step10: Solve the equation:
  $$\begin{align}&x=6\&x=-3\end{align}$$
- step11: Check if the solution is in the defined range:
  $$\begin{align}&x=6\&x=-3\end{align},x>5$$
- step12: Find the intersection:
  $$x=6$$
The solution to the equation $$ \log_{2}(x+2)+\log_{2}(x-5)=3 $$ is $$ x=6 $$.
$$ x=6 $$

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