Question

\[ 2^x = x^y = 12^z \]

Solution

Let \(2^x = x^y = 12^z = k\) \[ 2^x = k \Rightarrow x = \log_2 k \] \[ x^y = k \Rightarrow y = \frac{\log k}{\log x} \] \[ 12^z = k \Rightarrow z = \log_{12} k \] \[ \frac{1}{z} = \log_k 12 \] \[ = \log_k (2^2 \cdot 3) \] \[ = 2\log_k 2 + \log_k 3 \] \[ = \frac{2}{x} + \frac{1}{y} \]

Answer

\[ \boxed{\frac{1}{z} = \frac{1}{y} + \frac{2}{x}} \]

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