Question

Evaluate: 16^x – 4^x – 9 = 0.

(a) ln [( 5 + \(\sqrt{21}\)) / 2] / ln 8

(b) ln [( 2 + \(\sqrt{33}\)) / 2] / ln 5

(c) ln [( 1 + \(\sqrt{37}\)) / 2] / ln 4

(d) ln [( 1 – \(\sqrt{37}\)) / 2] / ln 3

This question was posed to me by my college professor while I was bunking the class.

My question comes from Discrete Probability topic in chapter Discrete Probability of Discrete Mathematics

Answer 1

Correct option is (c) ln [( 1 + \(\sqrt{37}\)) / 2] / ln 4

To elaborate: Given: 16^x – 4^x – 9 = 0. Since 16^x = (4^x)^2, the equation may be written as: (4^x)^2 – 4^x – 9 = 0. Let t = 3^x and so t: t^2 – t – 9 = 0 which gives t: t = (1 + \(\sqrt{37}\)) / 2 and (1 – \(\sqrt{37}\)) / 2

Since t = 4x, the acceptable solution is y = (1 + \(\sqrt{37}\)) / 2 ⇒ 4x = (1 + \(\sqrt{37}\))/2. By using ln on both sides: ln 4^x = ln [ (1 + \(\sqrt{37}\)) / 2]  ⇒ x = ln [ ( 1 + \(\sqrt{37}\))/2] / ln 3.