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Question Number 148048 by puissant last updated on 25/Jul/21

2^x −3^(((x/2))) =1

$$\mathrm{2}^{\mathrm{x}} −\mathrm{3}^{\left(\frac{\mathrm{x}}{\mathrm{2}}\right)} =\mathrm{1} \\ $$

Answered by mindispower last updated on 25/Jul/21

2^x −((√3))^x =1  ⇔1=((1/2))^x +(((√3)/2))^x   x=2  solution  if x≤0 ,((1/2))^x +(((√3)/2))^x ≥2  x→((1/2))^x +(((√3)/2))^x  decrease function for x>0  f(2)=1,⇒x=2

$$\mathrm{2}^{{x}} −\left(\sqrt{\mathrm{3}}\right)^{{x}} =\mathrm{1} \\ $$$$\Leftrightarrow\mathrm{1}=\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{{x}} +\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right)^{{x}} \\ $$$${x}=\mathrm{2}\:\:{solution} \\ $$$${if}\:{x}\leqslant\mathrm{0}\:,\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{{x}} +\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right)^{{x}} \geqslant\mathrm{2} \\ $$$${x}\rightarrow\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{{x}} +\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right)^{{x}} \:{decrease}\:{function}\:{for}\:{x}>\mathrm{0} \\ $$$${f}\left(\mathrm{2}\right)=\mathrm{1},\Rightarrow{x}=\mathrm{2} \\ $$$$ \\ $$$$ \\ $$

Answered by Rasheed.Sindhi last updated on 25/Jul/21

2^x −3^(((x/2))) =1  2^(2y) −3^y =1     [x/2→y]  4^y −3^y =1  y>1⇒4^y −3^y >1  y>1⇒x/2>1→No solution  y=1⇒x=2

$$\mathrm{2}^{\mathrm{x}} −\mathrm{3}^{\left(\frac{\mathrm{x}}{\mathrm{2}}\right)} =\mathrm{1} \\ $$$$\mathrm{2}^{\mathrm{2}{y}} −\mathrm{3}^{{y}} =\mathrm{1}\:\:\:\:\:\left[{x}/\mathrm{2}\rightarrow{y}\right] \\ $$$$\mathrm{4}^{{y}} −\mathrm{3}^{{y}} =\mathrm{1} \\ $$$${y}>\mathrm{1}\Rightarrow\mathrm{4}^{{y}} −\mathrm{3}^{{y}} >\mathrm{1} \\ $$$${y}>\mathrm{1}\Rightarrow{x}/\mathrm{2}>\mathrm{1}\rightarrow{No}\:{solution} \\ $$$${y}=\mathrm{1}\Rightarrow{x}=\mathrm{2} \\ $$

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