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Find the number of digits of the number \(2^{2^{22}} \).

Note by Subham Subian 5 months, 1 week 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 = \(10^{3010000}\)

is it the answer?

Or 3010000 digits? – Md Zuhair · 5 months ago

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TopNewestHere, \(2^{2^{22}}\) = 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 = \(10^{3010000}\)

is it the answer?

Or 3010000 digits? – Md Zuhair · 5 months ago

Log in to reply