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Question Number 157209 by aliyn last updated on 21/Oct/21

Commented by aliyn last updated on 21/Oct/21

$p r o v e t h i s ? ?$
Commented by cortano last updated on 21/Oct/21

$a^{\log_{a} b} = b^{\log_{a} a} = b$ $[\log_{a} a = 1]$
	Answered by puissant last updated on 21/Oct/21

	$a^{{l o g}_{a} b} = e^{{l o g}_{a} b \times l n a} = e^{\frac{l n b}{l n a} \times l n a} = e^{l n b} = b$
	Answered by mr W last updated on 21/Oct/21

	$s a y a^{\log_{a} b} = c$ $\Rightarrow \log_{a} b = \log_{a} c (d e f i n i t i o n)$ $\Rightarrow \log_{a} b - \log_{a} c = 0$ $\Rightarrow \log_{a} (\frac{b}{c}) = 0$ $\Rightarrow \frac{b}{c} = 1$ $\Rightarrow c = b$ $\Rightarrow a^{\log_{a} b} = c = b$
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