To solve the logarithmic equation

$\log(x+10) - \log(x+4) = \log x,$

we proceed as follows:

Step 1: Apply the Logarithmic Property

Using the property $\log a - \log b = \log\left(\frac{a}{b}\right)$, we rewrite the left-hand side:

$\log\left(\frac{x+10}{x+4}\right) = \log x.$

Step 2: Equate the Arguments

Since the logarithms are equal, their arguments must also be equal. Thus, we have:

$\frac{x+10}{x+4} = x.$

Step 3: Eliminate the Fraction

Multiply through by $x+4$ (assuming $x \neq -4$):

$x + 10 = x(x + 4).$

Step 4: Expand and Rearrange

Expand the right-hand side:

$x + 10 = x^2 + 4x.$

Rearrange the terms to form a standard quadratic equation:

$x^2 + 4x - x - 10 = 0 \quad \Rightarrow \quad x^2 + 3x - 10 = 0.$

Step 5: Solve the Quadratic Equation

Factorize the quadratic equation:

$x^2 + 3x - 10 = (x + 5)(x - 2) = 0.$

Thus, the solutions are:

$x = -5 \quad \text{or} \quad x = 2.$

Step 6: Check for Extraneous Solutions

Logarithmic equations require the arguments of the logarithms to be positive. Check each solution:

• For $x = -5$: The arguments $x+10 = 5$, $x+4 = -1$, and $\log x = \log(-5)$ are not valid because logarithms of negative numbers are undefined.
• For $x = 2$: The arguments $x+10 = 12$, $x+4 = 6$, and $\log x = \log(2)$ are valid.

Thus, the only solution is:

$\boxed{x = 2}.$

Would you like a more detailed explanation of any step?

Here are 5 questions to expand on this:

1. What are some properties of logarithms used to simplify equations?
2. How can you identify and avoid extraneous solutions in logarithmic equations?
3. Could this equation be solved graphically? If so, how?
4. What alternative methods can be used to solve quadratic equations like $x^2 + 3x - 10 = 0$?
5. What are some common errors to watch out for when solving logarithmic equations?

Tip: Always check the domain restrictions of logarithmic equations before finalizing the solution.