Wolfram alpha tells me that $a^{1/\ln a} = e$ (Symbolab tells me it the LHS cannot be simplified).
Can you help me show this equivalence?
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2$a^{b}=e^{b \ln a}$. – geetha290krm Jul 01 '22 at 09:46
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1ln(a^(1/ln(a)))=ln(a)/ln(a)=1 – Jul 01 '22 at 09:49
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2A little unfortunate that the question isn't good enough, but the post What is the nth root of a number contains an answer to your question. – Sarvesh Ravichandran Iyer Jul 02 '22 at 05:05
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The best way is to convert everything to the same base, and the natural base is $e$. We write $$ a^{1 / \ln(a)} = \exp(\ln(a^{1/\ln(a)})) = \exp\left( \frac{1}{\ln(a)} \cdot \ln(a) \right) = \exp(1) . $$ The key fact is that for any real $x$, we can always write $x = e^{\ln(x)}$ since $x \mapsto \exp(x)$ and $x \mapsto \ln(x)$ are inverses.
JKL
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