ABC26GN1988 · Surds and Indices

Let $3^{x}=5^{y}=45^{z}=k$. Then, $3=k^{\frac{1}{x}}, 5=k^{\frac{1}{y}}, 45=k^{\frac{1}{z}}$. $$\begin{aligned} & 45=3^{2} \times 5 \\ & \Leftrightarrow k^{\frac{1}{z}}=\left(k^{\frac{1}{x}}\right)^{2} \cdot\left(k^{\frac{1}{y}}\right)=k^{\frac{2}{x}} \cdot k^{\frac{1}{y}}=k^{\left(\frac{2}{x}+\frac{1}{y}\right)} \\ & \Leftrightarrow \quad \frac{1}{z}=\frac{2}{x}+\frac{1}{y} \Leftrightarrow \frac{2}{x}=\frac{1}{z}-\frac{1}{y} . \end{aligned}$$