Let's solve the equation given:

$\log_{10} x - \log_{10} \sqrt{x} = 2 \log_x 10$

First, let's simplify the left-hand side of the equation using logarithmic properties.

Recall that $\log_{10} \sqrt{x} = \log_{10} x^{1/2} = \frac{1}{2} \log_{10} x$. Therefore:

$\log_{10} x - \frac{1}{2} \log_{10} x = \frac{1}{2} \log_{10} x$

The left-hand side simplifies to:

$\frac{1}{2} \log_{10} x$

So the equation now is:

$\frac{1}{2} \log_{10} x = 2 \log_x 10$

Next, we use the change of base formula for logarithms. Recall that:

$\log_x 10 = \frac{\log_{10} 10}{\log_{10} x}$

Since $\log_{10} 10 = 1$, we have:

$\log_x 10 = \frac{1}{\log_{10} x}$

Thus, the equation becomes:

$\frac{1}{2} \log_{10} x = 2 \cdot \frac{1}{\log_{10} x}$

Multiplying both sides by $\log_{10} x$ to eliminate the fraction:

$\frac{1}{2} (\log_{10} x)^2 = 2$

To isolate $(\log_{10} x)^2$, multiply both sides by 2:

$(\log_{10} x)^2 = 4$

Now, take the square root of both sides:

$\log_{10} x = \pm 2$

This gives us two cases:

1. $\log_{10} x = 2$
2. $\log_{10} x = -2$

For the first case:

$\log_{10} x = 2 \implies x = 10^2 = 100$

For the second case:

$\log_{10} x = -2 \implies x = 10^{-2} = 0.01$

Therefore, the solutions for $x$ are:

$\boxed{100 \text{ and } 0.01}$

Would you like further details on this explanation, or do you have any questions? Here are some related questions for practice:

1. If $\log_{10} (2x) - \log_{10} x = 1$, what is the value of $x$?
2. Solve for $x$ if $\log_{10} (x^2) - \log_{10} (x) = \log_{10} (x^3) - \log_{10} (x^2)$.
3. If $\log_{10} x + \log_{10} (x + 1) = 1$, what is the value of $x$?
4. Solve for $x$ if $3 \log_{10} x = 2 \log_{10} 5$.
5. What is the value of $x$ if $\log_{10} (5x) = \log_{10} (25) - 2$?
6. If $\log_{10} x = \frac{1}{3} \log_{10} 27$, what is the value of $x$?
7. Solve for $x$ if $\log_{10} (x + 3) - \log_{10} x = 1$.
8. What is the value of $x$ if $\log_{10} (x^2 + 3x) = \log_{10} (x^2) + \log_{10} (3x)$?

Tip: Remember that logarithmic equations often require properties such as the change of base formula, the power rule, and the product or quotient rules for simplification.