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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. 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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. 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