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 $x = 5$.

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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.