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Algebra Difficulty 3.0 AMC 10/12 Prove it Canada

Determine all values of $x$ for which $2^{\log_{10}(x^2)}=3\left(2^{1+\log_{10} x}\right)+16$ .

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.