If (2.4)x = (0.24)y = 10z then show that \(\frac1x-\frac1z=\frac1y\)

1.	If (2.4)x = (0.24)y = 10z then show that \(\frac1x-\frac1z=\frac1y\)
Answer» (2.4)^x = 10z ⇒ 2.4 = \(10^{\frac{z}{x}}\) and (0.24)^y = 10^z ⇒ 0.24 = \(10^{\frac{z}{y}}\) ∴ \(\frac{2.4}{0.24}=\frac{10^{\frac{z}{x}}}{10^{\frac{z}{y}}} ⇒ 10 = 10^{\frac{z}{x}-\frac{z}{y}}\) ⇒ 1 = \(\frac{z}{x}-\frac{z}{y} = z\big(\frac{1}{x}-\frac{1}{y}\big).\) ⇒ \(\frac{1}{x}-\frac{1}{y}=\frac{1}{z} \,or\,\) \(\frac{1}{x}-\frac{1}{z}=\frac{1}{y}.\)

Answer»

(2.4)^x = 10z ⇒ 2.4 = \(10^{\frac{z}{x}}\)

and (0.24)^y = 10^z ⇒ 0.24 = \(10^{\frac{z}{y}}\)

∴ \(\frac{2.4}{0.24}=\frac{10^{\frac{z}{x}}}{10^{\frac{z}{y}}} ⇒ 10 = 10^{\frac{z}{x}-\frac{z}{y}}\)

⇒ 1 = \(\frac{z}{x}-\frac{z}{y} = z\big(\frac{1}{x}-\frac{1}{y}\big).\)

⇒ \(\frac{1}{x}-\frac{1}{y}=\frac{1}{z} \,or\,\) \(\frac{1}{x}-\frac{1}{z}=\frac{1}{y}.\)

Discussion