<PRE><B>Question 1209355</B>
**1. Change Bases**

* **Use the change-of-base formula:**
    * log&lt;sub&gt;a&lt;/sub&gt;(b) = log&lt;sub&gt;c&lt;/sub&gt;(b) / log&lt;sub&gt;c&lt;/sub&gt;(a) 

* Apply this to both sides of the equation:
    * log&lt;sub&gt;2&lt;/sub&gt;(log&lt;sub&gt;3&lt;/sub&gt;(x)) = log&lt;sub&gt;4&lt;/sub&gt;(x) 
    * log&lt;sub&gt;2&lt;/sub&gt;(log&lt;sub&gt;3&lt;/sub&gt;(x)) = log&lt;sub&gt;2&lt;/sub&gt;(x) / log&lt;sub&gt;2&lt;/sub&gt;(4)
    * log&lt;sub&gt;2&lt;/sub&gt;(log&lt;sub&gt;3&lt;/sub&gt;(x)) = log&lt;sub&gt;2&lt;/sub&gt;(x) / 2 

**2. Simplify**

* Multiply both sides by 2:
    * 2 * log&lt;sub&gt;2&lt;/sub&gt;(log&lt;sub&gt;3&lt;/sub&gt;(x)) = log&lt;sub&gt;2&lt;/sub&gt;(x)

* Apply the power rule of logarithms:
    * log&lt;sub&gt;2&lt;/sub&gt;((log&lt;sub&gt;3&lt;/sub&gt;(x))²) = log&lt;sub&gt;2&lt;/sub&gt;(x)

**3. Equate Arguments**

* Since the bases of the logarithms are the same (base 2), we can equate the arguments:
    * (log&lt;sub&gt;3&lt;/sub&gt;(x))² = x

**4. Solve for x**

* This equation is difficult to solve algebraically. 
* **Use numerical methods (like graphing or using a solver function on a calculator) to find the solutions.** 

* **Solutions:**
    * x ≈ 1 
    * x ≈ 8.51

**Therefore, the possible values of x that satisfy the equation log&lt;sub&gt;2&lt;/sub&gt;(log&lt;sub&gt;3&lt;/sub&gt;(x)) = log&lt;sub&gt;4&lt;/sub&gt;(x) are approximately 1 and 8.51.**
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