in the question above the first log is to the base of x

Thank you for the clarification. Let's revisit the equation with the correct base:

The equation given is:

\[
\log_{x} 4 + \log_{4} x = 2
\]

Let's solve this step by step.

### 1. Express the equation in terms of logarithms:
- The equation is already expressed in terms of logarithms. We'll start by isolating one of the logarithms to solve for \(x\).

### 2. Use the property of logarithms:
- We'll rewrite the equation as it stands:
  \[
  \log_{x} 4 + \log_{4} x = 2
  \]

### 3. Consider the property \(\log_a b = \frac{1}{\log_b a}\):
- Notice that the term \(\log_{4} x\) can be rewritten using the change of base formula, which states \(\log_{4} x = \frac{1}{\log_x 4}\). However, in this case, we'll try another approach.

### 4. Substitute \( \log_x 4 = y \) (where \(y = \log_x 4\)):
- So, \( y + \frac{1}{y} = 2 \).

### 5. Multiply through by \( y \) to eliminate the fraction:
- This gives:
  \[
  y^2 + 1 = 2y
  \]
- Rearranging gives us a quadratic equation:
  \[
  y^2 - 2y + 1 = 0
  \]

### 6. Solve the quadratic equation:
- This equation can be factored as:
  \[
  (y - 1)^2 = 0
  \]
- Therefore, \( y = 1 \).

### 7. Substitute back \( y = \log_x 4 \):
- Since \( y = \log_x 4 = 1 \), this implies:
  \[
  x^1 = 4
  \]
- Therefore, \( x = 4 \).

### Final Solution:
The value of \(x\) that satisfies the equation is \(x = 4\).

Would you like to explore any further details or ask any other questions?

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Here are some related questions:

1. How does the change of base formula work for logarithms?
2. Can you solve another equation involving logarithms with different bases?
3. What happens when the base of a logarithm is a variable?
4. How can you graph logarithmic functions with different bases?
5. What are some real-life applications of logarithmic functions?

**Tip:** Quadratic equations often appear when dealing with logarithms and exponents. Remember to check for possible factorization to simplify the solution process.

Learn how to solve the logarithmic equation log_x 4 + log_4 x = 2 step-by-step. This problem involves logarithms with different bases and requires the use of the change of base formula. Suitable for high school students in Grades 10-12.

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