Question

\[ a^x = b^y = c^z,\quad b^2 = ac \]

Solution

\[ \text{Let } a^x = b^y = c^z = k \] \[ a = k^{1/x},\quad b = k^{1/y},\quad c = k^{1/z} \] \[ b^2 = ac \Rightarrow (k^{1/y})^2 = k^{1/x} \cdot k^{1/z} \] \[ k^{2/y} = k^{1/x + 1/z} \] \[ \frac{2}{y} = \frac{1}{x} + \frac{1}{z} \] \[ \frac{2}{y} = \frac{z + x}{xz} \] \[ y = \frac{2xz}{x + z} \]

Answer

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

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