Why is it true that $a^{\log_{b}n} = n^{\log_{b}a}$
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What properties of logarithms do you know? What have you tried? – ViktorStein Nov 26 '19 at 21:41
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ye man, the guys below already explained it but thanks! :) – Peter Dec 05 '19 at 22:49
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$a^{\log_b n} = a^{\frac{\ln n}{\ln b}} = e^{\frac{\ln n}{\ln b}\ln a} = n^{\frac{\ln a}{\ln b}} = n^{\log_b a}$ for $a,b,n>0$.
Mister Da
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By definition
$$x={\log_{b}n} \iff b^x=n$$
$$y={\log_{b}a} \iff b^y=a$$
therefore
$$a^{\log_{b}n} =a^x=b^{xy}=n^y= n^{\log_{b}a}$$
user
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