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Question Number 208594 by Jimenez000 last updated on 18/Jun/24

$$Find$$$$x+3^x<4$$

Answered by mr W last updated on 18/Jun/24

$$3^x<4-x$$$$(4-x)3^{-x}>1$$$$(4-x)3^{4-x}>3^4$$$$(4-x)\ln 3e^{(4-x)\ln 3}>3^4\ln 3$$$$(4-x)\ln 3>W\left(3^4\ln 3\right)$$$$x<4-\frac{W\left(3^4\ln 3\right)}{\ln 3}=1$$

Commented by Jimenez000 last updated on 18/Jun/24

$$\mathrm{From}\mathrm{step}3\mathrm{to}4,\mathrm{what}\mathrm{exactly}\mathrm{did}\mathrm{do}\mathrm{you}?,\mathrm{i}{\mathrm{can}}^{\prime }\mathrm{t}\mathrm{see}\mathrm{it}$$

Commented by mr W last updated on 18/Jun/24

$$sayW\left(3^4\ln 3\right)=a$$$${ae}^a=3^4\ln 3=3^3\left(3\ln 3\right)=\left(3\ln 3\right)e^{\left(3\ln 3\right)}$$$$\Rightarrow a=3\ln 3$$$$4-\frac{W\left(3^4\ln 3\right)}{\ln 3}=4-\frac{3\ln 3}{\ln 3}=4-3=1$$

Commented by Jimenez000 last updated on 18/Jun/24

$$\mathrm{Thank}\mathrm{you}\mathrm{so}\mathrm{much}\mathrm{mr}\mathrm{W}$$

Commented by Kademi last updated on 18/Jun/24

$$$$$$W\left({xe}^x\right)=x$$$$W\left(a^c\ln a\right)=W\left(a^{a^b+b}\ln a\right)=W(a^b\ln a\cdot e^{a^b\ln a})=a^b\ln a$$

Answered by Jimenez000 last updated on 18/Jun/24

$$3^x<4-x$$$$x\ln 3<\ln (4-x)$$$$x<\frac{\ln (4-x)}{\ln 3}$$$$\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{know}\mathrm{how}\mathrm{to}\mathrm{continue}$$

Commented by Kademi last updated on 18/Jun/24

$$x<\frac{\ln (4-x)}{\ln 3}$$$$x<\frac{4-x}{3}$$$$3x<4-x$$$$4x-4<0$$$$4(x-1)<0$$$$\{\begin{array}{l}4\cancel{<}0\Rightarrow 4>0\\ x-1<0\Rightarrow x<1\end{array}$$

Answered by mr W last updated on 18/Jun/24

$$f\left(x\right)=x+3^x$$$$f^{\prime }\left(x\right)=1+3^x\ln 3>0\Rightarrow strictlyincreasing$$$$f\left(1\right)=1+3^1=4$$$$f\left(x\right)=x+3^x<4=f\left(1\right)$$$$\Rightarrow x<1$$

Answered by Kademi last updated on 18/Jun/24

$$$$$$3^x<4-x$$$$e^{x\ln 3}<4-x$$$$1<(4-x)e^{-x\ln 3}$$$$\ln 3e^{4\ln 3}<(4-x)\ln 3e^{4\ln 3-x\ln 3}$$$$\ln 3e^{\ln 3^4}<(4-x)\ln 3e^{(4-x)\ln 3}$$$$W\left(3^4\ln 3\right)<W\left((4-x)\ln 3e^{(4-x)\ln 3}\right)$$$$W\left(3^4\ln 3\right)<(4-x)\ln 3$$$$\frac{W\left(3^4\ln 3\right)}{\ln 3}<4-x$$$$x<4-\frac{W\left(3^4\ln 3\right)}{\ln 3}$$$$x<4-\frac{W\left(3\ln 3e^{3\ln 3}\right)}{\ln 3}$$$$x<4-\frac{3\ln 3}{\ln 3}$$$$x<4-3$$$$x<1$$$$$$$$$$

Commented by Jimenez000 last updated on 18/Jun/24

$$\mathrm{Thank}\mathrm{you}\mathrm{Kademi}$$

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