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Question Number 15672 by Mr easymsn last updated on 12/Jun/17

$$solvetheequation:$$$$\left\{5^x\right\}+5x=140$$$$pleaseshowworkings.....$$

Answered by mrW1 last updated on 13/Jun/17

$$5\mathrm{x}=140-\left\{5^{\mathrm{x}}\right\}⩽140\mathrm{and}>139$$$$\Rightarrow \mathrm{x}⩽140/5=28$$$$\Rightarrow \mathrm{x}>139/5=27.8$$$$\mathrm{for}\mathrm{x}=28:$$$$\left\{5^{28}\right\}+5\times 28=0+140=140\mathrm{ok}$$$$\mathrm{for}\mathrm{x}=27.\mathrm{xxx}=27+\mathrm{f}:$$$$\left\{5^{27+\mathrm{f}}\right\}+5\mathrm{f}=5$$$$\mathrm{f}=1-\frac{1}{5}\left\{5^{27}\times 5^{\mathrm{f}}\right\}$$$$\mathrm{there}\mathrm{must}\mathrm{exist}\mathrm{solutions}\mathrm{for}\mathrm{f}.$$$$$$$$\mathrm{so}\mathrm{one}\mathrm{solution}\mathrm{is}\mathrm{certain}:\mathrm{x}=28,$$$$\mathrm{other}\mathrm{solutions}\mathrm{x}=27+\mathrm{f}\mathrm{exist},\mathrm{but}$$$$\mathrm{can}\mathrm{not}\mathrm{be}\mathrm{calculated}.$$

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