Solve the following logarithmic equation algebraically.
\[
\log _{2}(x)+\log _{2}(x+2)=\log _{2} 8
\]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution is $x=$
(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers.)
B. The solution is not a real number.
Answer
Final Answer: The solution is \(x=\boxed{2}\).
Step 1 ：Combine the two logarithms on the left side of the equation into one using the property of logarithms that states that the sum of two logarithms with the same base is the logarithm of the product of the numbers. This gives us \(\log _{2}(x(x+2))=\log _{2} 8\).
Step 2 ：Use the property of logarithms that states that if two logarithms with the same base are equal, then their arguments are equal. This gives us \(x(x+2)=8\).
Step 3 ：Solve the quadratic equation \(x(x+2)=8\) for x. This gives us the solution \(x=2\).
Step 4 ：Final Answer: The solution is \(x=\boxed{2}\).
