Solve the following logarithmic equation.

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is { }.
(Simplify your answer. Type an exact answer. Use a comma to separate answers as needed.
B. There is no solution.

Ask by Wright Craig. in the United States

Dec 16,2024

Question

Solve the following logarithmic equation.

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is { }.
(Simplify your answer. Type an exact answer. Use a comma to separate answers as needed.
B. There is no solution.

Ask by Wright Craig. in the United States

Dec 16,2024

UpStudy AI · Accepted Answer

Solve the equation $$ \log_{2}(x+5)+\log_{2}(x+2)=2 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\log_{2}{\left(x+5ight)}+\log_{2}{\left(x+2ight)}=2$$
- step1: Find the domain:
  $$\log_{2}{\left(x+5ight)}+\log_{2}{\left(x+2ight)}=2,x>-2$$
- step2: Transform the expression:
  $$\log_{2}{\left(\left(x+5ight)\left(x+2ight)ight)}=2$$
- step3: Convert the logarithm into exponential form:
  $$\left(x+5ight)\left(x+2ight)=2^{2}$$
- step4: Evaluate the power:
  $$\left(x+5ight)\left(x+2ight)=4$$
- step5: Expand the expression:
  $$x^{2}+7x+10=4$$
- step6: Move the expression to the left side:
  $$x^{2}+7x+10-4=0$$
- step7: Subtract the numbers:
  $$x^{2}+7x+6=0$$
- step8: Factor the expression:
  $$\left(x+1ight)\left(x+6ight)=0$$
- step9: Separate into possible cases:
  $$\begin{align}&x+1=0\&x+6=0\end{align}$$
- step10: Solve the equation:
  $$\begin{align}&x=-1\&x=-6\end{align}$$
- step11: Check if the solution is in the defined range:
  $$\begin{align}&x=-1\&x=-6\end{align},x>-2$$
- step12: Find the intersection:
  $$x=-1$$
The solution to the logarithmic equation $$\log_{2}(x+5)+\log_{2}(x+2)=2$$ is $$x=-1$$.

Therefore, the correct choice is:
A. The solution set is \{ -1 \}.
The solution set is \{ -1 \}.

Solve the following logarithmic equation.

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is { }.
(Simplify your answer. Type an exact answer. Use a comma to separate answers as needed.
B. There is no solution.

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Solve the following logarithmic equation. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is { }. (Simplify your answer. Type an exact answer. Use a comma to separate answers as needed. B. There is no solution.

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Solve the following logarithmic equation.

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is { }.
(Simplify your answer. Type an exact answer. Use a comma to separate answers as needed.
B. There is no solution.