Question Number 26249 by Gabriel Wendell Celestino Roch last updated on 23/Dec/17
$$\mathrm{If}\:\:\:{x},\:{y},\:{z}\:\mathrm{are}\:\mathrm{in}\:\mathrm{GP}\:\mathrm{and}\:{a}^{{x}} =\:{b}^{{y}} =\:{c}^{{z}} ,\:\mathrm{then} \\ $$
Commented by Rasheed.Sindhi last updated on 23/Dec/17
![y=rx,z=r^2 x [r is common ratio] a^x =b^(rx) =c^(r^2 x) x log a=rx log b=r^2 x log c log a=r log b=r^2 log c r=((log a)/(log b)) =((log b)/(log c)) ∴ log a , log b & log c are also in GP having ratio 1/r resiprocal to the ratio in GP x,y & z](https://www.tinkutara.com/question/Q26272.png)
$$\mathrm{y}=\mathrm{rx},\mathrm{z}=\mathrm{r}^{\mathrm{2}} \mathrm{x}\:\left[\mathrm{r}\:\mathrm{is}\:\mathrm{common}\:\mathrm{ratio}\right] \\ $$$$\mathrm{a}^{\mathrm{x}} =\mathrm{b}^{\mathrm{rx}} =\mathrm{c}^{\mathrm{r}^{\mathrm{2}} \mathrm{x}} \\ $$$$\mathrm{x}\:\mathrm{log}\:\mathrm{a}=\mathrm{rx}\:\mathrm{log}\:\mathrm{b}=\mathrm{r}^{\mathrm{2}} \mathrm{x}\:\mathrm{log}\:\mathrm{c} \\ $$$$\:\mathrm{log}\:\mathrm{a}=\mathrm{r}\:\mathrm{log}\:\mathrm{b}=\mathrm{r}^{\mathrm{2}} \:\mathrm{log}\:\mathrm{c} \\ $$$$\mathrm{r}=\frac{\mathrm{log}\:\mathrm{a}}{\mathrm{log}\:\mathrm{b}}\:=\frac{\mathrm{log}\:\mathrm{b}}{\mathrm{log}\:\mathrm{c}} \\ $$$$\therefore\:\mathrm{log}\:\mathrm{a}\:,\:\mathrm{log}\:\mathrm{b}\:\&\:\mathrm{log}\:\mathrm{c}\:\mathrm{are}\:\mathrm{also} \\ $$$$\:\:\:\:\:\mathrm{in}\:\mathrm{GP}\:\mathrm{having}\:\:\mathrm{ratio}\:\mathrm{1}/\mathrm{r}\:\mathrm{resiprocal} \\ $$$$\:\:\:\mathrm{to}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{in}\:\mathrm{GP}\:\mathrm{x},\mathrm{y}\:\&\:\mathrm{z} \\ $$