If (2.4)^x = (0.24)^y = 10^z then show that (1/x)-(1/z)=(1/y)

Question

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

Answer 1

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

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