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If-y-a-1-1-log-a-x-and-z-a-1-1-log-a-y-show-that-x-a-1-1-log-a-z-




Question Number 42549 by Tawa1 last updated on 27/Aug/18
If   y = a^(1/(1 − log_a x))      and        z = a^(1/(1 − log_a y))   .  show that     x = a^(1/(1 − log_a z))
$$\mathrm{If}\:\:\:\mathrm{y}\:=\:\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{1}\:−\:\mathrm{log}_{\mathrm{a}} \mathrm{x}}} \:\:\:\:\:\mathrm{and}\:\:\:\:\:\:\:\:\mathrm{z}\:=\:\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{1}\:−\:\mathrm{log}_{\mathrm{a}} \mathrm{y}}} \:\:.\:\:\mathrm{show}\:\mathrm{that}\:\:\:\:\:\mathrm{x}\:=\:\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{1}\:−\:\mathrm{log}_{\mathrm{a}} \mathrm{z}}} \\ $$
Answered by math1967 last updated on 27/Aug/18
log_a y=(1/(1−log_a x))⇒log_a x=1−(1/(log_a y))  similarly  log_a y=1−(1/(log_a z))  ∴log_a x=1−(1/(1−(1/(log_a z))))=((−1)/(log_a z−1))=(1/(1−log_a z))  ∴x=a^(1/(1−log_a z))
$${log}_{{a}} {y}=\frac{\mathrm{1}}{\mathrm{1}−{log}_{{a}} {x}}\Rightarrow{log}_{{a}} {x}=\mathrm{1}−\frac{\mathrm{1}}{{log}_{{a}} {y}} \\ $$$${similarly}\:\:{log}_{{a}} {y}=\mathrm{1}−\frac{\mathrm{1}}{{log}_{{a}} {z}} \\ $$$$\therefore{log}_{{a}} {x}=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{1}−\frac{\mathrm{1}}{{log}_{{a}} {z}}}=\frac{−\mathrm{1}}{{log}_{{a}} {z}−\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{1}−{log}_{{a}} {z}} \\ $$$$\therefore{x}={a}^{\frac{\mathrm{1}}{\mathrm{1}−{log}_{{a}} {z}}} \\ $$
Commented by Tawa1 last updated on 27/Aug/18
God bless you sir
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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