find-the-value-of-y-given-that-y-y-3-y-9-

Question Number 207747 by MASANJAJJ last updated on 25/May/24

f i n d t h e v a l u e o f y g i v e n t h a t y^{y} = 3^{y + 9}

Commented by A5T last updated on 25/May/24

3^{9 + 9} = 3^{18} = {(3^{2})}^{3^{2}} = 9^{9} \Rightarrow y = 9

Answered by mr W last updated on 25/May/24

y^{y} = 3^{y} \times 3^{9}

{(\frac{y}{3})}^{y} = 3^{9}

{(\frac{y}{3})}^{\frac{y}{3}} = 3^{3}

\Rightarrow \frac{y}{3} = 3 \Rightarrow y = 9

Answered by Berbere last updated on 25/May/24

y^{y} = 3^{a} \Rightarrow a = \frac{y l n (y)}{l n (3)}

\Leftrightarrow 3^{\frac{y l n (y)}{l n (3)}} = 3^{y + 9}

\Rightarrow y l n (y) = y l n (3) + 9 l n (3)

y l n (\frac{y}{3}) = 9 l n (3) > 0 \Rightarrow y ⩾ 3

y \overset{f}{\to} y l n (\frac{y}{3}) i n c r e a s e f u n c t i o n o v e r [3, \infty [

f^{'} = l n (\frac{y}{3}) + \frac{y}{3} ⩾ l n (\frac{3}{3}) + \frac{3}{3} = 1

f (9) = 9 l n (3) \Rightarrow y = 9

Answered by Berbere last updated on 25/May/24

i f y \in N^{*}

\Rightarrow y = 3^{a}; a ⩾ 1

3^{a y} = 3^{y + 9}

a y = y + 9

y ∣ a y \Leftrightarrow y ∣ y + 9 \Rightarrow y ∣ (a y - y) = 9 \Rightarrow y ∣ 9

\Rightarrow y \in {1, 3, 9} \Rightarrow y = 9 b y t c h e k i n g

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