Question

\[ 3^x = 5^y = 75^z \]

Solution

\[ \text{Let } 3^x = 5^y = 75^z = k \] \[ 3^x = k \Rightarrow x = \log_3 k,\quad 5^y = k \Rightarrow y = \log_5 k \] \[ 75^z = k \Rightarrow (3 \cdot 5^2)^z = k \] \[ 3^z \cdot 5^{2z} = k \] \[ 3^z = k^{1} \cdot 5^{-2z} \] \[ \Rightarrow z = \frac{xy}{2x + y} \]

Answer

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

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