Let a , b , c be the three distinct positive real numbers such that 2 a log e a = b c log e b and b log e 2 = a log e c Then 6 a + 5 b c is equal to ______.

Step-by-step solution

Given equation: 2 a ln a = b c ln b ⇒ ln a · ln 2 a = ln b · ln b c ⇒ ln a · ln 2 + ln a = ln b · ln b + ln c let ln a = x ,   ln b = y ,   ln c = z ,   x ≠ y ≠ z ⇒ x ln 2 + x = y y + z ⇒ x ln 2 = y 2 - x 2 + y z ........(i) Similarly, from second equation ln 2 · ln b = ln c · ln a ⇒ ln 2 = x z y ......(ii) Substitute e q i i   in   e q i , We get, ⇒ y 3 - y x 2 + y 2 z = x 2 z ⇒ y 2 y + z - x 2 y + z = 0 &#