Solve the equation. Write the solution set with the exact solutions.

There is no solution,
The exact solution set is
Th.

Ask by Wagner Clark. in the United States

Mar 25,2025

Question

Solve the equation. Write the solution set with the exact solutions. $$ \log _{3} w+\log _{3}(w+6)=3 $$ There is no solution, $$ \} $$ The exact solution set is Th.

UpStudy AI · Accepted Answer

Solve the equation $$ \log _{3} w+\log _{3}(w+6)=3 $$.
Solve the equation by following steps:
- step0: Solve for $$w$$:
  $$\log_{3}{\left(wight)}+\log_{3}{\left(w+6ight)}=3$$
- step1: Find the domain:
  $$\log_{3}{\left(wight)}+\log_{3}{\left(w+6ight)}=3,w\gt 0$$
- step2: Transform the expression:
  $$\log_{3}{\left(w\left(w+6ight)ight)}=3$$
- step3: Convert the logarithm into exponential form:
  $$w\left(w+6ight)=3^{3}$$
- step4: Evaluate the power:
  $$w\left(w+6ight)=27$$
- step5: Expand the expression:
  $$w^{2}+6w=27$$
- step6: Move the expression to the left side:
  $$w^{2}+6w-27=0$$
- step7: Factor the expression:
  $$\left(w-3ight)\left(w+9ight)=0$$
- step8: Separate into possible cases:
  $$\begin{align}&w-3=0\&w+9=0\end{align}$$
- step9: Solve the equation:
  $$\begin{align}&w=3\&w=-9\end{align}$$
- step10: Check if the solution is in the defined range:
  $$\begin{align}&w=3\&w=-9\end{align},w\gt 0$$
- step11: Find the intersection:
  $$w=3$$
The solution to the equation $$ \log _{3} w+\log _{3}(w+6)=3 $$ is $$ w=3 $$.
The exact solution set is $$ w=3 $$.

Solve the equation. Write the solution set with the exact solutions.

There is no solution,
The exact solution set is
Th.

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Solve the equation. Write the solution set with the exact solutions. There is no solution, The exact solution set is Th.