To solve the equation $\log x + \log (x + 3) = 1$, we can use the properties of logarithms. Specifically, we use the property that states $\log a + \log b = \log (ab)$.

1. Combine the logarithms: $\log (x(x + 3)) = 1$

2. Exponentiate both sides to eliminate the logarithm: $x(x + 3) = 10^1$ $x(x + 3) = 10$

3. Expand the left side: $x^2 + 3x - 10 = 0$

4. Now, we can solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1, b = 3, c = -10$.

5. Calculate the discriminant: $b^2 - 4ac = 3^2 - 4(1)(-10) = 9 + 40 = 49$

6. Substitute into the quadratic formula: $x = \frac{-3 \pm \sqrt{49}}{2(1)} = \frac{-3 \pm 7}{2}$

7. This gives us two solutions: $x = \frac{4}{2} = 2 \quad \text{and} \quad x = \frac{-10}{2} = -5$

Since $x$ must be positive in the context of logarithms, we discard $x = -5$.

Thus, the solution is: $\boxed{2}$

Would you like details on any step?

Here are some related questions you might consider:

1. How do logarithmic properties help in solving equations?
2. What are the conditions under which a logarithmic function is defined?
3. Can you provide examples of other equations involving logarithms?
4. What happens if we consider $\log x + \log (x + 3) = k$ for other values of $k$?
5. How can quadratic equations be solved using factoring?

Tip: Always check your solutions in the original equation, especially when dealing with logarithms, to avoid extraneous solutions.