Question

How do you solve `log _(2) (log _(2) (log _(2) x))) = 2`?

Answer 1

Make repeated use of the fact that `2^(log_2(x)) = x` to find that
`x = 65536`

Explanation:

Applying the property of logarithms that

`a^(log_a(x)) = x`

we have

`log_2(log_2(log_2(x))) = 2`

`= 2^(log_2(log_2(log_2(x)))) = 2^2`

`=> log_2(log_2(x)) = 4`

`=> 2^(log_2(log_2(x))) = 2^4`

`=> log_2(x) = 16`

`=> 2^(log_2(x)) = 2^16`

`=> x = 65536`

Answer 2

`x=2^16=65536`

Explanation:

Start from this point: if you know that `a=b`, then it must be `2^a=2^b`. So, start from your equation, and deduce that

`2^{log_2(log_2(log_2(x)))}=2^2`

Now, by definition, `2^{log_2(z)} = z`, while of course `2^2=4`. So, the equation becomes

`log_2(log_2(x))=4`

And now we're in the same situation as before, so we can apply the same logic:

`2^{log_2(log_2(x))}=2^4`

Which means

`log_2(x)=16`

One last iteration gives us

`x=2^16=65536`

Verify the answer:

`log_2(log_2(log_2(2^16))) = log_2(log_2(16))=log_2(4)=2`