Going way too up

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

#NumberTheory

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
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Comments

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 - 4 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$