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Question Number 42019 by Tawa1 last updated on 16/Aug/18

$$\mathrm{find}\mathrm{x}:2^{\mathrm{x}}+3^{\mathrm{x}}=13$$

Commented by math khazana by abdo last updated on 17/Aug/18

$$letf\left(x\right)=2^x+3^x-13\left(e\right)\iff f\left(x\right)=0$$$$f\left(x\right)=e^{xln\left(2\right)}+e^{xln3}-13\Rightarrow$$$$f^{{}^{\prime }}\left(x\right)=ln\left(2\right)2^x+ln\left(3\right)3^x>0sofisincreasing$$$$onRwehave{lim}_{x\to -\mathrm{\infty }}f\left(x\right)=-13and$$$${lim}_{x\to +\mathrm{\infty }}f\left(x\right)=+\mathrm{\infty }so\mathrm{\exists }{x}_0\in R/f\left(x_0\right)=0$$$$wehavef\left(0\right)=-13<0andf\left(3\right)=2^3+3^3-13$$$$=8+27-13=22>0\Rightarrow x_0\in ]0,3[wecan$$$$verifythat{x}_0=2isexactsolutiontof\left(x\right)=0$$

Commented by Tawa1 last updated on 17/Aug/18

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

Commented by maxmathsup by imad last updated on 17/Aug/18

$$thankssir$$

Answered by MJS last updated on 17/Aug/18

$$x=2$$

Commented by Tawa1 last updated on 17/Aug/18

$$\mathrm{yes}\mathrm{sir}.\mathrm{but}\mathrm{any}\mathrm{workings}??$$

Commented by MJS last updated on 17/Aug/18

$$2^x+3^x=13$$$$x=\frac{\ln t}{\ln 2}$$$$t^{\frac{\ln 3}{\ln 2}}+t=13$$$$\mathrm{since}\frac{\ln 3}{\ln 2}\mathrm{is}\mathrm{irrational}\mathrm{we}\mathrm{can}\mathrm{only}\mathrm{approximate}$$

Commented by Tawa1 last updated on 17/Aug/18

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

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