Question:

<div class="solution"><p><strong>Key Idea:</strong> Collapse the tower using $a^{\log_a t} = t$ and reciprocal log identity.</p><p><strong>Step 1:</strong> Domain: $\log_x 2012 > 0$ requires $x > 1$.</p><p><strong>Step 2:</strong> <span class="math-block">$$f_3(x) = \left(\frac{1}{2013}\right)^{\log_{2013}(\log_x 2012)} = \frac{1}{\log_x 2012} = \log_{2012} x$$</p><p><strong>Step 3:</strong> $f_2(x) = 2012^{\log_{2012} x} = x$</p><p><strong>Step 4:</strong> $f_1(x) = 2^x$, $x > 1$, so range is $(2,\infty)$</p><p><strong>Answer: $(2,\infty)$</strong></p><div class="trap-box"><strong>Trap:</strong> Domain is critical -- $\log_x 2012$ must also be positive (not just defined) because it is an argument of another log.<div class="key-concept"><strong>Key Concept:</strong> Nested exponent-log telescoping using $a^{\log_a t}=t$