Log the log

Algebra Level 3

Given that $\large 2^{x} = x^{2^{65520}}$ , find the value of $\log_{2}({\log_{2}{x}})$ .

1 solution

Jeremy Galvagni
Apr 8, 2018

Taking the base-2 logarithm of both sides of the original equation gives

$x = 2^{65520}\log_{2}x$

doing so again gives

$\log_{2}x = 65520 + \log_{2}(\log_{2}x)$

Note that $2^{16}=65536 = 65520+16$

And letting $\log_{2}(\log_{2}x)=16$ so that $\log_{2}x=65536$ we get a perfect solution so indeed

$\boxed{\log_{2}(\log_{2}x)=16}$

Judging from your solution I gather you meant the equation in the question to be

$\Large 2^{x} = x^{2^{65520}}$ .

Brian Charlesworth - 3 years, 2 months ago

thanks. still getting the hang of latex

Jeremy Galvagni - 3 years, 2 months ago