Using the root adjunction formalism developed in an earlier work and logarithmic THH, we obtain a simplified computation of $T(2)_*\text{K}(ku)$ for $p>3$. Through this, we also produce a new algebraic $K$-theory computation; namely we obtain $T(2)_*\text{K}(ku/p)$, where $ku/p$ is the $2$-periodic Morava $K$-theory spectrum of height $1$.
