if-log-9-a-log-4-a-b-log-6-b-what-is-a-b-

Question Number 77219 by john santu last updated on 04/Jan/20

i f \log_{9} (a) = \log_{4} (a + b) = \log_{6} (b)

w h a t i s \frac{a}{b} ?

Answered by jagoll last updated on 04/Jan/20

suppose \log_{9} (a) = t .

\Rightarrow a = 9^{t}, a + b = 4^{t} and b = 6^{t}

so we written \Rightarrow 9^{t} + 6^{t} = 4^{t}

\frac{9^{t}}{6^{t}} + 1 = \frac{4^{t}}{6^{t}} \Rightarrow {(\frac{3}{2})}^{t} - {(\frac{2}{3})}^{t} + 1 = 0

let {(\frac{3}{2})}^{t} = x \Rightarrow x - \frac{1}{x} + 1 = 0

x^{2} + x - 1 = 0 \Rightarrow x = \frac{- 1 + \sqrt{5}}{2}

so we get x = \frac{3^{t}}{2^{t}} = {(\frac{3 \times 3}{3 \times 2})}^{t} = \frac{9^{t}}{6^{t}} = \frac{\sqrt{5} - 1}{2} = \frac{a}{b}

({it}^{'} s namely^{'} The Golden {Ratio}^{'}

Commented by jagoll last updated on 04/Jan/20

\approx 0.6180

Commented by john santu last updated on 05/Jan/20

t h a n k s y o u