Question 1209355: Find x if \log_2 (\log_3 x) = \log_4 x.
Answer by yurtman(42)   (Show Source):
You can put this solution on YOUR website! **1. Change Bases**
* **Use the change-of-base formula:**
* loga(b) = logc(b) / logc(a)
* Apply this to both sides of the equation:
* log2(log3(x)) = log4(x)
* log2(log3(x)) = log2(x) / log2(4)
* log2(log3(x)) = log2(x) / 2
**2. Simplify**
* Multiply both sides by 2:
* 2 * log2(log3(x)) = log2(x)
* Apply the power rule of logarithms:
* log2((log3(x))²) = log2(x)
**3. Equate Arguments**
* Since the bases of the logarithms are the same (base 2), we can equate the arguments:
* (log3(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 log2(log3(x)) = log4(x) are approximately 1 and 8.51.**
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