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

2 Answers
Dec 17, 2015

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

Dec 17, 2015

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