a^(lna) =b^(lnb) a−b=1 solve for a and b


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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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