Question 9 Solve the equation: $$ \log _{4}(x+2)-\log _{4}(x-3)=2 $$

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Solve the equation $$ \log _{4}(x+2)-\log _{4}(x-3)=2 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\log_{4}{\left(x+2ight)}-\log_{4}{\left(x-3ight)}=2$$
- step1: Find the domain:
  $$\log_{4}{\left(x+2ight)}-\log_{4}{\left(x-3ight)}=2,x>3$$
- step2: Transform the expression:
  $$\log_{4}{\left(\frac{x+2}{x-3}ight)}=2$$
- step3: Convert the logarithm into exponential form:
  $$\frac{x+2}{x-3}=4^{2}$$
- step4: Evaluate the power:
  $$\frac{x+2}{x-3}=16$$
- step5: Cross multiply:
  $$x+2=\left(x-3ight)	imes 16$$
- step6: Simplify the equation:
  $$x+2=16\left(x-3ight)$$
- step7: Expand the expression:
  $$x+2=16x-48$$
- step8: Move the expression to the left side:
  $$x-16x=-48-2$$
- step9: Add and subtract:
  $$-15x=-48-2$$
- step10: Add and subtract:
  $$-15x=-50$$
- step11: Change the signs:
  $$15x=50$$
- step12: Divide both sides:
  $$\frac{15x}{15}=\frac{50}{15}$$
- step13: Divide the numbers:
  $$x=\frac{10}{3}$$
- step14: Check if the solution is in the defined range:
  $$x=\frac{10}{3},x>3$$
- step15: Find the intersection:
  $$x=\frac{10}{3}$$
The solution to the equation $$ \log _{4}(x+2)-\log _{4}(x-3)=2 $$ is $$ x = \frac{10}{3} $$ or $$ x = 3.\dot{3} $$.
$$ x = \frac{10}{3} $$ or $$ x = 3.\dot{3} $$

Question 9
Solve the equation:

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