a-lna-b-lnb-a-b-1-solve-for-a-and-b-

Question Number 176310 by Linton last updated on 16/Sep/22

a^{l n a} = b^{l n b}

a - b = 1

s o l v e f o r a a n d b

Answered by mr W last updated on 16/Sep/22

{(\ln a)}^{2} = {(\ln b)}^{2}

\Rightarrow \ln a = \pm \ln b

\Rightarrow a = b (r e j e c t e d d u e t o a - b = 1)

o r a = \frac{1}{b}

\frac{1}{b} - b = 1

b^{2} + b - 1 = 0

\Rightarrow b = \frac{- 1 + \sqrt{5}}{2} > 0

\Rightarrow a = \frac{1}{b} = \frac{\sqrt{5} + 1}{2}

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