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Question Number 142503 by Gbenga last updated on 01/Jun/21

Answered by mr W last updated on 01/Jun/21

$$x^{x^{40}}={50}^x$$$$x^{x^{39}}=50$$$${\left(x^{x^{39}}\right)}^{39}={50}^{39}$$$${\left(x^{39}\right)}^{x^{39}}={50}^{39}$$$$lett=x^{39}$$$$t^t={59}^{39}$$$$\ln t=\frac{39\ln 50}{t}$$$$39\ln 50=\frac{39\ln 50}{t}{e}^{\frac{39\ln 50}{t}}$$$$\frac{39\ln 50}{t}=W\left(39\ln 50\right)$$$$t=\frac{39\ln 50}{W\left(39\ln 50\right)}$$$$x^{39}=t=\frac{39\ln 50}{W\left(39\ln 50\right)}$$$$\Rightarrow x=\sqrt[39]{\frac{39\ln 50}{W\left(39\ln 50\right)}}$$$$\approx \sqrt[39]{\frac{39\ln 50}{3.71518721}}=1.099946$$

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