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How to solve $10x=3^x+3$? There should be two answers from the graph plotted, one should be 3, but how do I get the other one?

jjagmath
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min yu
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    There is no algebraic method that solves this equation. – Ethan Bolker Nov 15 '23 at 00:29
  • But why is there no algebraic method? – min yu Nov 15 '23 at 00:33
  • The solutions are $3$ and $$\frac{3}{10}-\frac{W\left(-\tfrac{1}{10}3^{3/10}\log(3)\right)}{\log 3}$$ where $W$ is the Lambert function. – jjagmath Nov 15 '23 at 00:41
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    Asking why there is no algebraic method is both a naive and a good question. Most equations you can write down can't be solved by the kinds of manipulations you learn when you first learn algebra. Proving that some can and some can't is often difficult. The comment from @jjagmath uses the Lambert function. That was "invented" precisely to name the solutions to this kind of equation. – Ethan Bolker Nov 15 '23 at 00:45

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