log-5-x-9-log-5-3x-2-12-log-5-2x-1-0-

Question Number 156109 by cortano last updated on 08/Oct/21

\log_{5} (\sqrt{x - 9}) - \log_{5} ({3 x}^{2} - 12) - \log_{5} (\sqrt{2 x - 1}) ⩽ 0

Answered by yeti123 last updated on 08/Oct/21

(1) \cap (2) \cap (3) \cap (4) \cap (5) \cap (6) = 9 < x ⩽ 21478

Answered by yeti123 last updated on 08/Oct/21

\log_{5} (\sqrt{x - 9}) - \log_{5} (3 x^{2} - 12) - \log_{5} (\sqrt{2 x - 1}) ⩽ 0

\log_{5} (\frac{\sqrt{x - 9}}{(3 x^{2} - 12) (\sqrt{2 x - 1})}) ⩽ \log_{5} (1)

\frac{\sqrt{x - 9}}{(3 x^{2} - 12) \sqrt{2 x - 1}} ⩽ 1

\Rightarrow x = 9 \dots . (1)

- 2 < x < \frac{1}{2} \dots . . (2)

x ⩽ 21478 \dots \dots . . (3)

\log_{5} (\sqrt{x - 9}) \Rightarrow \sqrt{x - 9} > 0 \Rightarrow x > 9 \dots \dots \dots \dots . . (4)

\log_{5} (3 x^{2} - 12) \Rightarrow (3 x^{2} - 12) > 0 \Rightarrow x > 2 \land x < - 2 \dots \dots \dots . (5)

\log_{5} (\sqrt{2 x - 1}) \Rightarrow \sqrt{2 x - 1} > 0 \Rightarrow x > \frac{1}{2} \dots \dots \dots \dots (6)

(1) \cap (2) \cap (3) \cap (4) \cap (5) \cap (6) = \frac{1}{2} < x ⩽ 21478

Commented by cortano last updated on 08/Oct/21

for x = 1 is not solution

Commented by mr W last updated on 08/Oct/21

w h y x ⩽ 21478 ? h o w d i d y o u g e t i t ?

i n f a c t w e h a v e x > 9 a n d f o r a n y x > 9

0 < \frac{\sqrt{x - 9}}{(3 x^{2} - 12) \sqrt{2 x - 1}} ⩽ 0.0009 ⩽ 1

s o t h e a n s w e r i s x \in (9, + \infty)

Commented by yeti123 last updated on 08/Oct/21

yes mr . W, I miscalculated . thanks for the correction .

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