  Given an $E_1$-ring $A$ and a class $a \in \pi_{mk}(A)$ satisfying a suitable
hypothesis, we define a map of $E_1$-rings $A\to A(\sqrt[m]{a})$ realizing the
adjunction of an $m$th root of $a$. 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$(A(\sqrt[m]{a}))$
in terms of THH and log-THH of $A$. If $A$ is connective, we prove that the
induced map $K(A) \to K(A(\sqrt[m]{a}))$ 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 $K(A)$ exhibits chromatic redshift, so does
$K(A(\sqrt[m]{a}))$. We interpret several extensions of ring spectra as
examples of root adjunction, and use this to obtain a new proof of the fact
that Lubin-Tate spectra satisfy the redshift conjecture.
