A Long Product of Logarithms

Solve for the value of $k$ such that $$ \log_2 3 \times \log_3 4 \times \log_4 5 \times \dots \times \log_{k-1} k = 2026 $$The answer is in the form $\alpha^\beta $ where $\alpha$ is a prime number. Input the answer as $\alpha + \beta$.