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Question Number 82665 by jagoll last updated on 23/Feb/20

$$\sqrt{6561\sqrt{6561\sqrt{6561\sqrt{6561\sqrt{...}}}}}=3^{8^x}$$$$\sqrt{6561\sqrt{...}}\left(60time\right)$$$$findx$$

Commented by mr W last updated on 23/Feb/20

$${6561}^{\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{60}}}=3^{8^x}$$$${6561}^{\frac{1}{2}\left(\frac{1-\frac{1}{2^{60}}}{1-\frac{1}{2}}\right)}=3^{8^x}$$$${6561}^{1-\frac{1}{2^{60}}}=3^{8^x}$$$$3^{8(1-\frac{1}{2^{60}})}=3^{8^x}$$$$8(1-\frac{1}{2^{60}})=8^x$$$$1-\frac{1}{2^{60}}=8^{x-1}=2^{3(x-1)}$$$$\Rightarrow x=1+\frac{1}{3}{\log }_2(1-\frac{1}{2^{60}})$$

Commented by jagoll last updated on 23/Feb/20

$$thankyoumisterwandmisterjohn$$

Answered by john santu last updated on 23/Feb/20

$$3^{\frac{4}{2}+\frac{4}{4}+\frac{4}{8}+\frac{4}{16}+...+\frac{4}{2^{60}}}=3^{8^x}$$$$\Rightarrow 4(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{2^{60}})=8^x$$$$\Rightarrow 4\left(\frac{\left(\frac{1}{2}\right)(1-{\left(\frac{1}{2}\right)}^{60})}{1-\frac{1}{2}}\right)=8^x$$$$\Rightarrow 1-2^{-60}=2^{3x-2}$$$$\Rightarrow 3x-2={\log }_2(1-2^{-60})$$$$\Rightarrow x=\frac{{\log }_2(4-2^{-58})}{3}$$

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