2 Many Logs?

Algebra Level 3

$\large\log_2 ( \log_2 (\log_2 (\log_2 (x)))) = 2$

If $x = 2^n$ satisfy the equation above, what is the value of $n$ ?

The answer is 65536.

2 solutions

Michael Fuller
Jun 3, 2015

If $\large\log _{ \color{#3D99F6}{2} }{ \color{#D61F06}a } =\color{#20A900}{2}$ , then $\large \color{#D61F06}a=\color{#3D99F6}{ 2 }^\color{#20A900}{ 2 }$ - the base of the log 'bumps' up. Therefore we have:

$\large \Rightarrow x=2\uparrow \uparrow 5={ 2 }^{ 2^{ { 2 }^{ { 2 }^{ 2 } } } }$ (where $2\uparrow \uparrow 5$ denotes a tetration )

$\large \Rightarrow n=2\uparrow \uparrow 4=2^{ { 2 }^{ { 2 }^{ 2 } } }$

$\large n=\boxed { 65536 }$

Moderator note:

Thank you for incorporating tetration notation into your solution.

Andrew Zakharov
Jun 2, 2015

$\large \log_2 ( \log_2 (\log_2 (\log_2 (X)))) = 2$ $\large \log_2 (\log_2 (\log_2 (X))) = 4$ $\large \log_2 (\log_2 (X)) = 16$ $\large \log_2 (N) = 16$ N=65536

Moderator note:

Good standard approach.

the solution can be writeen as X= 2^16 so here n = 16. why is the answer 2^16 insted of just 16. they have just asked for n

Gokul Kumar - 5 years, 10 months ago

If u refer to the last line, log2(N) = 16, this means that N = 2^16.

Owen Leong - 5 years, 7 months ago

2 Many Logs?

The answer is 65536.

2 solutions

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