Question-136520

Question Number 136520 by mey3nipaba last updated on 22/Mar/21

Commented by mey3nipaba last updated on 22/Mar/21

help please .

Answered by mindispower last updated on 22/Mar/21

\Leftrightarrow {\begin{cases} 2^{x} = 3^{y} \Rightarrow x = \frac{y l n (3)}{l n (2)} \\ (x - 1) = \frac{y - 1}{l n (3)} l n (2) \end{cases}

Answered by Rasheed.Sindhi last updated on 23/Mar/21

2^{x + y} = 6^{y} \land 3^{x} = 3 (2^{y - 1})

2^{x} = 3^{y} \dots . (i) \land 3^{x - 1} = 2^{y - 1} \dots . (i i)

(i) \times (i i) :

2^{x + y - 1} = 3^{x + y - 1}

\Rightarrow x + y - 1 = 0

x + y = 1 \dots \dots \dots \dots \dots \dots (i i i)

(i) / (i i) :

\frac{2^{x}}{2^{y - 1}} = \frac{3^{y}}{3^{x - 1}}

2^{x - y + 1} = 3^{y - x + 1}

x - y + 1 = 0 \land y - x + 1 = 0

x - y + 1 = y - x + 1

x - y = y - x

x = y \dots \dots \dots \dots \dots \dots \dots \dots . . (i v)

(i i i) & (i v) :

x = y = 1 / 2

{D o e s n}^{'} t s a t i s f y o r i g i n a l e q u a t i o n .

∴ N o s o l u t i o n .

Commented by bemath last updated on 23/Mar/21

i f x = y = \frac{1}{2} \Rightarrow 2^{x + y} = 2^{\frac{1}{2} + \frac{1}{2}} = 2^{1}

R H S \Rightarrow 6^{y} = 6^{\frac{1}{2}} = \sqrt{6}

w h y 2 \neq \sqrt{6} s i r ?

Commented by Rasheed.Sindhi last updated on 23/Mar/21

Y e s m a d a m, p e r h a p s {t h e r e}^{'} s n o s o l u t i o n .

Commented by mr W last updated on 23/Mar/21

3^{a} = 2^{b} ⇏ a = b = 0

i t h i n k {i t}^{'} s w r o n g h e r e :

2^{x - y + 1} = 3^{y - x + 1}

⇏ x - y + 1 = 0 \land y - x + 1 = 0

\Rightarrow x - y + 1 = (y - x + 1) \log_{2} 3

\Rightarrow x - y = \frac{\log_{2} 3}{1 + \log_{2} 3}

Commented by Rasheed.Sindhi last updated on 23/Mar/21

T h α n k s f o r g u i d l i n e s i r!

Answered by mr W last updated on 23/Mar/21

2^{x} 2^{y} = 2^{y} 3^{y}

2^{x} = 3^{y}

\Rightarrow x = y \log_{2} 3 = k y

w i t h k = \log_{2} 3

3^{x - 1} = 2^{y - 1}

y - 1 = (x - 1) \log_{2} 3 = (x - 1) k

y - 1 = (k y - 1) k

\Rightarrow y = \frac{1 - k}{1 - k^{2}} = \frac{1}{1 + k} = \frac{1}{1 + \log_{2} 3} \approx 0.386853

\Rightarrow x = \frac{k}{1 + k} = \frac{1}{1 + \frac{1}{k}} = \frac{1}{1 + \log_{3} 2} \approx 0.613147

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