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To solve the equation

\[
\log x + \log (x + 3) = 1,
\]

we can proceed with the following steps:

1. **Combine the logarithmic terms:**  
   Using the property \(\log a + \log b = \log (a \cdot b)\), we rewrite the left side as a single logarithm:

   \[
   \log (x \cdot (x + 3)) = 1.
   \]

   Simplifying inside the logarithm:

   \[
   \log (x^2 + 3x) = 1.
   \]

2. **Rewrite in exponential form:**  
   Since \(\log (x^2 + 3x) = 1\), we can rewrite this equation in exponential form as:

   \[
   x^2 + 3x = 10^1.
   \]

   Simplifying the right side:

   \[
   x^2 + 3x = 10.
   \]

3. **Rewrite as a standard quadratic equation:**  
   Move all terms to one side to form a quadratic equation:

   \[
   x^2 + 3x - 10 = 0.
   \]

4. **Solve the quadratic equation:**  
   We can use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 3\), and \(c = -10\):

   \[
   x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-10)}}{2 \cdot 1}.
   \]

   Simplify under the square root:

   \[
   x = \frac{-3 \pm \sqrt{9 + 40}}{2}.
   \]

   \[
   x = \frac{-3 \pm \sqrt{49}}{2}.
   \]

   \[
   x = \frac{-3 \pm 7}{2}.
   \]

   This gives two solutions:

   \[
   x = \frac{-3 + 7}{2} = 2 \quad \text{and} \quad x = \frac{-3 - 7}{2} = -5.
   \]

5. **Check for valid solutions:**  
   Since \(x = -5\) would make \(\log x\) undefined (we cannot take the logarithm of a negative number), we discard \(x = -5\). Thus, the only valid solution is:

   \[
   x = 2.
   \]

So, the solution set is:
\[
\{2\}.
\]

Would you like further details or have any questions?

---

Here are 5 related questions for further practice:

1. Solve \(\log (x + 2) + \log (x + 4) = 2\).
2. If \(\log x + \log (x - 5) = 2\), find \(x\).
3. Solve the equation \(\log (2x) + \log (x + 1) = 1\).
4. Determine the solution to \(\log (x) + \log (x + 6) = 2\).
5. Solve \(\log (x + 3) + \log (x + 5) = 1\).

**Tip:** Always check if the values satisfy the domain of the logarithmic function, especially in cases where the solution includes negative or zero values.

Solve the following logarithmic equation: log x + log (x + 3) = 1

Learn how to solve the logarithmic equation log x + log (x + 3) = 1 with a detailed step-by-step solution. This problem covers key concepts in logarithms and quadratic equations, including the properties of logarithms and the quadratic formula. Suitable for students in Grades 10-12.

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