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# Going way too up

Find the number of digits of the number $$2^{2^{22}}$$.

Note by Subham Subian
2 months ago

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Here, $$2^{2^{22}}$$ = x , Taking $$log_{10}$$ on both sides we get $$2^{22} log _{10}2 = log_{10}x$$

Now $$2^{22} = y$$

$$log_{10}y = 22 log_{10}2$$

Hence log 2 = 0.301 hence $$log _{10} y = 22 * 0.301$$

y = 10^{6.62}

Y has 7 digits in it.

Now $$y \lesssim 10^{7}$$

Hence Now $$10^7 x 0.301$$ = $$log_{10}x$$

$$3010000$$ = $$log_{10}x hence x = \(10^{3010000}$$