Solve for the variable. $$ \log _{3} x+\log _{3}(x+2)=1 $$ 1 no solution .5 3

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UpStudy AI · Accepted Answer

Solve the equation $$ \log_{3}(x) + \log_{3}(x+2) = 1 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\log_{3}{\left(xight)}+\log_{3}{\left(x+2ight)}=1$$
- step1: Find the domain:
  $$\log_{3}{\left(xight)}+\log_{3}{\left(x+2ight)}=1,x>0$$
- step2: Transform the expression:
  $$\log_{3}{\left(x\left(x+2ight)ight)}=1$$
- step3: Convert the logarithm into exponential form:
  $$x\left(x+2ight)=3^{1}$$
- step4: Evaluate the power:
  $$x\left(x+2ight)=3$$
- step5: Expand the expression:
  $$x^{2}+2x=3$$
- step6: Move the expression to the left side:
  $$x^{2}+2x-3=0$$
- step7: Factor the expression:
  $$\left(x-1ight)\left(x+3ight)=0$$
- step8: Separate into possible cases:
  $$\begin{align}&x-1=0\&x+3=0\end{align}$$
- step9: Solve the equation:
  $$\begin{align}&x=1\&x=-3\end{align}$$
- step10: Check if the solution is in the defined range:
  $$\begin{align}&x=1\&x=-3\end{align},x>0$$
- step11: Find the intersection:
  $$x=1$$
The solution to the equation $$ \log_{3}x + \log_{3}(x+2) = 1 $$ is $$ x = 1 $$.
$$ x = 1 $$

Solve for the variable.

1
no solution
.5
3

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Solve for the variable. 1 no solution .5 3