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This came up in a differential equation, and I wondered if there is an algebraic way to solve this for $t$ without using WolframAlpha. Or is it a case of estimating with a power series?

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    Specifically, $t = 10 + 10 W(-54/(49 e))$ where $W$ is any of the branches of the Lambert $W$ function. However, none of the solutions is real.2012-11-18

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Numerical analysis will make short work of this, but don't expect an algebraic solution. Polynomials and exponentials don't play nice together. Any one dimensional root finder should handle it easily.

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Set $t=10z$. Then you have to solve: $\frac{49}{54}(1-z)=e^{-z},$ that has no real solutions in virtue of the convexity of the RHS.

In fact, for any positive real number $r$, $\forall z\in\mathbb{R},\quad e^{-z}\geq \frac{1}{r}(\log(r)+1-z)$ holds, so, if we choose $r=\frac{54}{49}$, we have: $e^{-z}\geq\frac{49}{54}\left(\log(54/49)+1-z\right)>\frac{49}{54}(1-z).$