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# Logg'e' function to the base $$x$$

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

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

4 months, 2 weeks ago

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$y= \log_{x}e^{\log_{x}e^{.... \text{50 times}}}=\log^{50}_{x}e=\frac{1}{\ln^{50}x} \\ \frac{dy}{dx}=-\frac{50\ln^{49}x\cdot \frac{1}{x} }{\ln ^{100}x}=-\frac{50}{x\ln^{51}x} \\ \frac{dy}{dx}|_{x=2}=-\frac{25}{\ln^{51}2}$ · 4 months, 2 weeks ago