Logg'e' function to the base xx

If y=logxelogxe....50 timesy= \log_{x}e^{\log_{x}e^{.... \text{50 times}}}

Then find dydx\dfrac{dy}{dx} at x=2x=2

Note by A Former Brilliant Member
3 years, 5 months ago

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y=logxelogxe....50 times=logx50e=1ln50xdydx=50ln49x1xln100x=50xln51xdydxx=2=25ln512y= \log_{x}e^{\log_{x}e^{.... \text{50 times}}}=\log^{50}_{x}e=\frac{1}{\ln^{50}x} \\ \frac{dy}{dx}=-\frac{50\ln^{49}x\cdot \frac{1}{x} }{\ln ^{100}x}=-\frac{50}{x\ln^{51}x} \\ \frac{dy}{dx}|_{x=2}=-\frac{25}{\ln^{51}2}

Akshat Sharda - 3 years, 5 months ago

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