The given equation is:

\[
\log^2 4 \cdot x \cdot x = 8
\]

This expression is a bit ambiguous due to the formatting, but I will assume the intended equation is:

\[
(\log_2 4)^2 \cdot x^2 = 8
\]

### Step-by-step solution:

1. **Simplify the logarithm**:
   \[
   \log_2 4 = 2 \quad \text{(since } 2^2 = 4\text{)}
   \]
   
   Now the equation becomes:
   \[
   2^2 \cdot x^2 = 8
   \]

2. **Simplify the squared values**:
   \[
   4 \cdot x^2 = 8
   \]

3. **Solve for \( x^2 \)**:
   Divide both sides by 4:
   \[
   x^2 = \frac{8}{4} = 2
   \]

4. **Solve for \( x \)**:
   Take the square root of both sides:
   \[
   x = \pm \sqrt{2}
   \]

### Final Answer:
\[
x = \pm \sqrt{2}
\]

Would you like further details on any part of the solution? Here are some related questions you could explore:

1. What are the properties of logarithms?
2. How can you solve logarithmic equations involving different bases?
3. What is the general method for solving quadratic equations?
4. How does the square root function behave with positive and negative values?
5. Can complex numbers arise in logarithmic or exponential problems?

**Tip**: When dealing with equations, always clarify ambiguous expressions to avoid mistakes.

This solution explains how to solve the logarithmic and quadratic equation (log2 4)^2 * x^2 = 8 step-by-step. It covers the simplification of logarithms and solving quadratic equations, making it perfect for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation