Given an $E_1$-ring $X$ and a class $x \in \pi_{mk}(X)$ satisfying a suitable hypothesis, we define a map of $E_1$-rings $X\to X(\sqrt[m]{x})$ realizing the adjunction of an $m$th root of $x$. We define a form of logarithmic THH for $E_1$-rings, and show that root adjunction is log-THH-\'etale for suitably tamely ramified extension, which provides a formula for THH$(X(\sqrt[m]{x}))$ in terms of THH and log-THH of $X$. If $X$ is connective, we prove that the induced map $K(X) \to K(X(\sqrt[m]{x}))$ in algebraic $K$-theory is the inclusion of a wedge summand. Using this, we obtain $V(1)_*K(ko_p)$ for $p>3$ and also, we deduce that if $K(X)$ exhibits chromatic redshift, so does $K(X(\sqrt[m]{x}))$. We interpret several extensions of ring spectra as examples of root adjunction, and use this to establish redshift in many new and known examples, including Morava $K$-theory $K(n)$, all $E_1$ $MU$-algebra forms of $BP \langle n \rangle$, and various quotients of $MU$.