If log10x – log10√x = 2 logx10, then solve for x:

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.

Learn how to solve the logarithmic equation log10x - log10√x = 2 logx10 step-by-step with detailed explanations. This problem involves logarithmic properties and the change of base formula. Suitable for high school students.

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