Solve the logarithmic equation $$ \log _{5}(x+10)+\log _{5}(x-10)=3 $$

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Solve the equation $$ \log_{5}(x+10)+\log_{5}(x-10)=3 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\log_{5}{\left(x+10ight)}+\log_{5}{\left(x-10ight)}=3$$
- step1: Find the domain:
  $$\log_{5}{\left(x+10ight)}+\log_{5}{\left(x-10ight)}=3,x>10$$
- step2: Transform the expression:
  $$\log_{5}{\left(\left(x+10ight)\left(x-10ight)ight)}=3$$
- step3: Convert the logarithm into exponential form:
  $$\left(x+10ight)\left(x-10ight)=5^{3}$$
- step4: Evaluate the power:
  $$\left(x+10ight)\left(x-10ight)=125$$
- step5: Expand the expression:
  $$x^{2}-100=125$$
- step6: Move the constant to the right side:
  $$x^{2}=125+100$$
- step7: Add the numbers:
  $$x^{2}=225$$
- step8: Simplify the expression:
  $$x=\pm \sqrt{225}$$
- step9: Simplify:
  $$x=\pm 15$$
- step10: Separate into possible cases:
  $$\begin{align}&x=15\&x=-15\end{align}$$
- step11: Check if the solution is in the defined range:
  $$\begin{align}&x=15\&x=-15\end{align},x>10$$
- step12: Find the intersection:
  $$x=15$$
The solution to the logarithmic equation $$ \log _{5}(x+10)+\log _{5}(x-10)=3 $$ is $$ x=15 $$.
$$ x=15 $$.

Solve the logarithmic equation

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