https://file.aiquickdraw.com/mathbot/file/1726594573199bkrykrw7.jpg

The given equation is:

\[
\lg x^2 - 4 \lg x + 3 = 0
\]

### Step-by-Step Solution:

1. **Use logarithmic property**: Recall that \(\lg x^2 = 2 \lg x\) by the power rule of logarithms.

   Therefore, the equation becomes:
   \[
   2 \lg x - 4 \lg x + 3 = 0
   \]

2. **Combine like terms**: 
   \[
   (2 - 4)\lg x + 3 = 0
   \]
   \[
   -2 \lg x + 3 = 0
   \]

3. **Isolate \(\lg x\)**:
   \[
   -2 \lg x = -3
   \]
   \[
   \lg x = \frac{3}{2}
   \]

4. **Solve for \(x\)**: Use the definition of logarithms, \(\lg x = y\) means \(x = 10^y\). So, 
   \[
   x = 10^{\frac{3}{2}} = 10^{1.5}
   \]

   This simplifies to:
   \[
   x = \sqrt{100} = 10\sqrt{10}
   \]

Thus, the solution is:
\[
x = 10^{1.5}
\]

Would you like more details on this solution? Here are 5 related questions for further exploration:

1. How does the power rule of logarithms apply to logarithmic equations?
2. Can we solve similar logarithmic equations with different bases?
3. What are the common properties of logarithms, and how can they simplify complex equations?
4. How does the solution of logarithmic equations relate to exponential equations?
5. What happens if the equation contains multiple logarithmic terms with different bases?

**Tip**: When solving logarithmic equations, always remember to check if the solutions make sense within the domain of the logarithm (i.e., \(x > 0\)).

Solve the equation lg x^2 - 4lg x + 3 = 0.

Step-by-step guide to solving the logarithmic equation lg x^2 - 4lg x + 3 = 0 using logarithmic properties and exponentiation. Learn about the power rule of logarithms and how to apply it to solve complex equations. Suitable for high school students (Grades 9-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation