Determine-in-simplest-form-the-smallest-of-the-three-numbers-x-y-and-z-which-satisfy-the-system-log-9-x-log-9-y-log-3-z-2-log-16-x-log-4-y-log-16-z-1-log-5-x-lo

Question Number 115859 by bemath last updated on 29/Sep/20

D e t e r m i n e, i n s i m p l e s t f o r m t h e

s m a l l e s t o f t h e t h r e e n u m b e r s x,

y a n d z w h i c h s a t i s f y t h e s y s t e m

{\begin{cases} \log_{9} (x) + \log_{9} (y) + \log_{3} (z) = 2 \\ \log_{16} (x) + \log_{4} (y) + \log_{16} (z) = 1 \\ \log_{5} (x) + \log_{25} (y) + \log_{25} (z) = 0 \end{cases}

Answered by bobhans last updated on 29/Sep/20

\to {\begin{cases} \log_{9} ({x y z}^{2}) = 2 \to {x y z}^{2} = 81 \\ \log_{16} ({x y}^{2} z) = 1 \to {x y}^{2} z = 16 \\ \log_{25} (x^{2} y z) = 0 \to x^{2} y z = 1 \end{cases}

\Leftrightarrow {(x y z)}^{4} = 81 \times 16 \times 1 = {(6)}^{4}

\Rightarrow x y z = 6 {\begin{cases} z = \frac{81}{6} = \frac{27}{2} \\ y = \frac{16}{6} = \frac{8}{3} \\ x = \frac{1}{6} \end{cases}

Answered by floor(10²Eta[1]) last updated on 29/Sep/20

(I) \frac{\log_{3} (x)}{2} + \frac{\log_{3} (y)}{2} + \log_{3} (z) = 2

\log_{3} (x) + \log_{3} (y) + {2 \log}_{3} (z) = 4

\log_{3} ({xyz}^{2}) = 4

{xyz}^{2} = 81

(II) \log_{4} (x) + {2 \log}_{4} (y) + \log_{4} (z) = 2

\log_{4} ({xy}^{2} z) = 2

{xy}^{2} z = 16

(III) {2 \log}_{5} (x) + \log_{5} (y) + \log_{5} (z) = 0

\log_{5} (x^{2} yz) = 0

x^{2} yz = 1

{\begin{cases} {xyz}^{2} = 81 \\ {xy}^{2} z = 16 \\ x^{2} yz = 1 \end{cases}

xyz = \frac{81}{z}

\frac{81 y}{z} = 16 \Rightarrow 16 z = 81 y \Rightarrow 16 x = y

\frac{81 x}{z} = 1 \Rightarrow z = 81 x

x . 16 x . 81 x = \frac{81}{81 x} \Rightarrow x^{4} = \frac{1}{6^{4}} \Rightarrow x = \frac{1}{6}

y = \frac{8}{3}

z = \frac{27}{2}