Q. Positive integers ' $a$ ' and 'b' satisfy the condition $\log _2\left(\log _{2^{ a }}\left(\log _{2^{ b }}\left(2^{1000}\right)\right)\right)=0$. Find the sum of all possible values of ' $a$ '.
Continuity and Differentiability
Solution:
$ \log _{2^{ a }}\left(\log _{2^{ b }}\left(2^{1000}\right)\right)=1$
$\log _{2^b}\left(2^{1000}\right)=2^a \Rightarrow 2^{1000}=\left(2^b\right)^{2^a}=2^{b \cdot 2^a}$
Hence, $b \cdot 2^{ a }=1000$
