Solve the logarithmic equation. Be sure to reject any value of $x$ that is not in the domain of the original logarithmic expressions. Give the exact answer.
$$\log x+\log (x+2)=\log 15$$

Solve the equation to find the solution set. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Step 1 ：The given equation is a logarithmic equation. The properties of logarithms state that the sum of two logarithms with the same base is the logarithm of the product of the numbers. Therefore, we can combine the two logarithms on the left side of the equation into one logarithm.

Step 2 ：Using the property that if ${\log }_b(a)={\log }_b(c)$, then $a=c$ to solve for $x$.

Step 3 ：We find that the solution to the equation is $x=3$.

Step 4 ：We need to check if this solution is in the domain of the original logarithmic expressions. The domain of a logarithmic function is all positive real numbers.

Step 5 ：The solution $x=3$ is valid because it is a positive real number and it makes both $x$ and $x+2$ positive.

Step 6 ：Final Answer: The solution to the equation is $\boxed{{\displaystyle 3}}$.