New user? Sign up

Existing user? Sign in

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

Note by Subham Subian 9 months ago

Sort by:

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?

Log in to reply

Problem Loading...

Note Loading...

Set Loading...

## Comments

Sort by:

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?

Log in to reply