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Algebra Difficulty 3.0 AMC 10/12 Prove it Canada

Determine all points $(x, y)$ where the two curves $y = \log_{10}(x^4)$ and $y = (\log_{10} x)^3$ intersect.

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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.