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Question Number 40083 by tawa tawa last updated on 15/Jul/18

$$\mathrm{solve}\mathrm{for}\mathrm{x}:5^{\mathrm{x}}+5\mathrm{x}=140$$

Commented by MrW3 last updated on 16/Jul/18

$$5^x=(28-x)\times 5$$$$5^{x-28}=(28-x)\times 5^{-27}$$$$e^{(x-28)\ln 5}=(28-x)\times 5^{-27}$$$$5^{27}\times \ln 5=(28-x)\ln 5e^{(28-x)\ln 5}$$$$\Rightarrow (28-x)\ln 5=\mathbb{W}\left(5^{27}\ln 5\right)$$$$\Rightarrow x=28-\frac{\mathbb{W}\left(5^{27}\ln 5\right)}{\ln 5}=28-\frac{40.2359}{\ln 5}=3$$

Commented by tawa tawa last updated on 16/Jul/18

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

Answered by MJS last updated on 15/Jul/18

$$5^3+5\times 3=140$$

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