https://file.aiquickdraw.com/mathbot/file/1725981684636gfbby2z4.jpg

Let's break down the steps provided in the image for solving the logarithmic equation:

### Problem:  
Solve the equation:

\[
\log x + \log (x - 1) = \log (4x)
\]

### Step 1: Combine the logarithms  
Using the property of logarithms that \(\log a + \log b = \log (a \cdot b)\), we combine the left-hand side:

\[
\log (x(x - 1)) = \log (4x)
\]

### Step 2: Set the arguments equal  
Since the logarithms are equal, we can set their arguments equal to each other:

\[
x(x - 1) = 4x
\]

### Step 3: Expand and simplify  
Expand the left-hand side:

\[
x^2 - x = 4x
\]

Move all terms to one side:

\[
x^2 - x - 4x = 0
\]

This simplifies to:

\[
x^2 - 5x = 0
\]

### Step 4: Factor the equation  
Factor the quadratic equation:

\[
x(x - 5) = 0
\]

### Step 5: Solve for \(x\)  
Set each factor equal to 0:

\[
x = 0 \quad \text{or} \quad x = 5
\]

### Step 6: Check for valid solutions  
Since logarithms are not defined for non-positive arguments, \(x = 0\) is not valid because \(\log(0)\) is undefined. Therefore, the only solution is:

\[
x = 5
\]

### Summary:
The only valid solution is \(x = 5\).

---

Do you have any questions or need further details? Here are some related questions to explore this concept further:

1. Why can't logarithms take non-positive values as their argument?
2. How does the property \(\log a + \log b = \log (a \cdot b)\) work, and when is it applicable?
3. What would happen if the logarithms had different bases in this equation?
4. Can you explain how to solve logarithmic equations using different methods like graphing?
5. How would you solve this equation if the logarithm function included a constant offset, such as \(\log (x + 1)\)?

**Tip**: Always check the domain of the logarithmic functions before solving to ensure that your solutions are valid within the defined range.

This step-by-step solution explains how to solve the logarithmic equation log x + log(x - 1) = log(4x) by combining logarithms, setting the arguments equal, and solving the resulting quadratic equation. This problem involves key algebraic concepts such as logarithmic properties and the quadratic formula, making it ideal for students in Grades 10-12.

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