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Question Number 91671 by  M±th+et+s last updated on 02/May/20
show that  (x)^(1/(ln(x))) =e
$${show}\:{that} \\ $$$$\sqrt[{{ln}\left({x}\right)}]{{x}}={e} \\ $$
Commented by mr W last updated on 02/May/20
(x)^(1/(ln(x))) =x^(1/(ln x)) =e^(ln (x^(1/(ln x)) )) =e^((1/(ln x))×ln x) =e^1 =e
$$\sqrt[{{ln}\left({x}\right)}]{{x}}={x}^{\frac{\mathrm{1}}{\mathrm{ln}\:{x}}} ={e}^{\mathrm{ln}\:\left({x}^{\frac{\mathrm{1}}{\mathrm{ln}\:{x}}} \right)} ={e}^{\frac{\mathrm{1}}{\mathrm{ln}\:{x}}×\mathrm{ln}\:{x}} ={e}^{\mathrm{1}} ={e} \\ $$
Commented by  M±th+et+s last updated on 02/May/20
nice work thank you
$${nice}\:{work}\:{thank}\:{you} \\ $$

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