if (2.3)^x = (0.23)^y =1000 show that 1/x - 1/y = 1/3

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Please find below the solution to the asked query:

$$\begin{gathered} {\left(2.3\right)}^{\mathrm{x}}={\left( 0.23\right)}^{\mathrm{y}} = 1000 \\ \mathrm{x} = {\log }_{2.3}\left(1000\right) =\frac{ \log \left(1000 \right)}{ \log \left(2.3\right)} \\ \mathrm{y} = {\log }_{0.23}\left(1000\right) = \frac{ \log \left(1000 \right)}{ \log \left(0.23\right)} \\ \mathrm{Let}\text{'}\mathrm{s} \mathrm{use} \mathrm{base} 10 \mathrm{for} \mathrm{those} \mathrm{logs}, \mathrm{so} \mathrm{we} \mathrm{have}: \\ \mathrm{x} = \frac{3}{ \log \left(2.3\right)} \\ \mathrm{y} = \frac{3}{ \log \left(0.23\right)} \\ \frac{1}{\mathrm{x}} =\frac{\log \left(2.3\right)}{ 3} \\ \frac{1}{\mathrm{y}}= \frac{\log \left(0.23\right)}{ 3} \\ \frac{1}{\mathrm{x}} -\frac{1}{\mathrm{y}} = \frac{\left(\log \left(2.3 \right)- \log \left(0.23\right)\right)}{3} \\ \frac{1}{\mathrm{x}} -\frac{1}{\mathrm{y}}= \frac{\log \left({\displaystyle \frac{2.3}{0.23}}\right)}{3} \\ \frac{1}{\mathrm{x}} -\frac{1}{\mathrm{y}} = \frac{\log \left({\displaystyle 10}\right)}{3} \\ \mathbf{ }\frac{\mathbf{1}}{\mathbf{x}}\mathbf{ }\mathbf{-}\frac{\mathbf{1}}{\mathbf{y}}\mathbf{=}\mathbf{ }\frac{\mathbf{1}}{\mathbf{3}} \end{gathered}$$

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