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.