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Question Number 193721 by sciencestudentW last updated on 18/Jun/23

$p r o v e t h a t a^{{l o g}_{a} N} = N$
	Answered by MATHEMATICSAM last updated on 18/Jun/23

	$From \log definition we know if$ $a^{x} = N then we can write it as$ $x = \log_{a} N$ $a^{x} = N$ $\Rightarrow a^{\log_{a} N} = N [∵ x = \log_{a} N]$
	Answered by mr W last updated on 18/Jun/23

	$s a y a^{\log_{a} N} = M$ $f r o m d e f i n i t i o n w e h a v e$ $\log_{a} M = \log_{a} N .$ $s i n c e \log_{a} x i s o n e t o o n e f u n c t i o n, i . e .$ $\log_{a} x_{1} = \log_{a} x_{2} \Leftrightarrow x_{1} = x_{2}, w e g e t$ $M = N . t h a t m e a n s$ $a^{\log_{a} N} = N .$
	Answered by JDamian last updated on 18/Jun/23

	$f (x) = a^{x}$ $g (x) = \log_{a} (x)$ $g (x) = f^{- 1} (x)$ $f (x) = g^{- 1} (x)$ $g \circ f = f (g (x)) = x$ $f \circ g = g (f (x)) = x$ $f (g (N)) = N a^{\log_{a} N} = N$ $g (f (N)) = N \log_{a} (a^{N}) = N$
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