Given-for-x-y-z-gt-0-2-x-3-y-5-z-Arrange-2x-3y-5z-in-increasing-order-

Question Number 79978 by mr W last updated on 29/Jan/20

G i v e n f o r x, y, z > 0 :

2^{x} = 3^{y} = 5^{z}

A r r a n g e 2 x, 3 y, 5 z i n i n c r e a s i n g o r d e r .

Answered by mind is power last updated on 29/Jan/20

x l n (2) = y l n (3) = z l n (5)

y = \frac{x l n (2)}{l n (3)}

z = \frac{x l n (2)}{l n (5)}

5 z = \frac{5}{l n (5)} . l n (2) x

3 y = \frac{3 l n (2)}{l n (3)} x

l n (8) < l n (9) \Rightarrow

3 l n (2) < 2 l n (3)

\Rightarrow \frac{3 l n (2)}{l n (3)} < 2

\Rightarrow 3 y = \frac{3 l n (2)}{l n (3)} x < 2 x

\frac{5 l n (2)}{l n (5)} > 2 \Leftrightarrow l n (32) > l n (25) T r u e \Rightarrow

5 z = \frac{5 l n (2)}{l n (5)} x > 2 x \Rightarrow 5 z > 2 x > 3 y

Answered by key of knowledge last updated on 29/Jan/20

2^{x} = 3^{y} = 5^{z} \Rightarrow x . \log_{3} 2 = y = z . \log_{3} 5

2 x = 2 (\frac{y}{{1 og}_{3} 2}) = y . {2 \log}_{2} 3 \approx 3 . 16 y

5 z = \dots = z \times {5 \log}_{5} 3 \approx 3 . 41 y

\Rightarrow 3 y < 2 x < 5 z

Answered by mr W last updated on 30/Jan/20

2^{x} = 3^{y} = 5^{z} = t > 0

x \ln 2 = y \ln 3 = z \ln 5 = \ln t = s

\Rightarrow x = \frac{s}{\ln 2}

\Rightarrow 2 x = \frac{s}{\frac{1}{2} \ln 2} = \frac{s}{\ln \sqrt{2}}

\Rightarrow \frac{1}{2 x} = \frac{\ln \sqrt{2}}{s}

s i m i l a r l y

\Rightarrow \frac{1}{3 y} = \frac{\ln \sqrt[3]{3}}{s}

\Rightarrow \frac{1}{5 z} = \frac{\ln \sqrt[5]{5}}{s}

n o w w e o n l y n e e d t o c o m p a r e t h e

n u m b e r s \sqrt{2}, \sqrt[3]{3}, \sqrt[5]{5} .

3^{2} = 9 > 8 = 2^{3}

\Rightarrow 3 > {(\sqrt{2})}^{3}

\Rightarrow \sqrt[3]{3} > \sqrt{2}

2^{5} = 32 > 25 = 5^{2}

\Rightarrow 2 > {(\sqrt[5]{5})}^{2}

\Rightarrow \sqrt{2} > \sqrt[5]{5}

\Rightarrow \sqrt[3]{3} > \sqrt{2} > \sqrt[5]{5}

\Rightarrow \frac{1}{3 y} > \frac{1}{2 x} > \frac{1}{5 z}

\Rightarrow 3 y < 2 x < 5 z

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