https://file.aiquickdraw.com/mathbot/file/17263907766902f3a8v86.jpg

The equation shown in the image is:

\[
\log_{10} x + \log_{10} (x - 3) = \log_{10} 18
\]

Let's solve this step-by-step.

### Step 1: Use the logarithmic product rule
The logarithmic product rule states that:
\[
\log_b a + \log_b c = \log_b (a \cdot c)
\]
So, apply this to the left-hand side of the equation:
\[
\log_{10} (x(x - 3)) = \log_{10} 18
\]

### Step 2: Remove the logarithms
Since the logarithms have the same base (base 10), we can equate the arguments of the logarithms:
\[
x(x - 3) = 18
\]

### Step 3: Expand and solve the quadratic equation
Expand the left-hand side:
\[
x^2 - 3x = 18
\]

Move everything to one side:
\[
x^2 - 3x - 18 = 0
\]

### Step 4: Solve using the quadratic formula
The quadratic formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \(x^2 - 3x - 18 = 0\), \(a = 1\), \(b = -3\), and \(c = -18\). Substituting these values into the quadratic formula:
\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-18)}}{2(1)}
\]
\[
x = \frac{3 \pm \sqrt{9 + 72}}{2}
\]
\[
x = \frac{3 \pm \sqrt{81}}{2}
\]
\[
x = \frac{3 \pm 9}{2}
\]

### Step 5: Find the two possible values for \(x\)
\[
x = \frac{3 + 9}{2} = \frac{12}{2} = 6
\]
\[
x = \frac{3 - 9}{2} = \frac{-6}{2} = -3
\]

### Step 6: Check for extraneous solutions
Since the logarithm is undefined for non-positive arguments, we must reject any negative solutions. The term \(\log_{10} x\) and \(\log_{10} (x - 3)\) must both have positive arguments. Therefore, \(x - 3 > 0\), meaning \(x > 3\).

Thus, the only valid solution is:
\[
x = 6
\]

### Final Answer:
The solution is \(x = 6\).

Would you like more details or further explanation on any step?

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Here are five related questions to expand on this topic:
1. What is the general form of the logarithmic product rule?
2. How do you handle cases where logarithmic equations involve negative numbers?
3. Can you use logarithmic properties to solve exponential equations?
4. How would you solve a logarithmic equation if the bases were different?
5. What are the common logarithmic laws that simplify complex logarithmic expressions?

**Tip:** Always check for extraneous solutions when solving logarithmic equations, especially when dealing with negative values in the domain.

log₁₀ x + log₁₀ (x - 3) = log₁₀ 18

Learn how to solve the logarithmic equation log₁₀ x + log₁₀ (x - 3) = log₁₀ 18 step-by-step using the logarithmic product rule and quadratic formula. This problem involves logarithmic and algebraic concepts suitable for students in Grades 10-12.

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