Algebraic $K$-theory of the two-periodic first Morava $K$-theory

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