perplexus.info :: Just Math : log the log

log the log (Posted on 2006-06-11)

Given that

2^x = x^{(2⁶⁵⁵²⁰)}

find log₂(log₂(x)).

See The Solution Submitted by Jer

Rating: 3.5000 (4 votes)

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re: Willing to bet

| Comment 2 of 5 |

(In reply to Willing to bet by e.g.)

Indeed, by taking the base-2 log of each side we get

x = 2^65520 * log2(x)

and then doing the same again

log2(x) = 65520 + log2(log2(x))

If x=2^(2^16), then log2(x) = 2^16 = 65536 and log2(log2(x)) = 16, and the equation is satisfied, so

log2(log2(x)) = 16

Sweet.

Posted by Charlie on 2006-06-11 11:36:08

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