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Question-32709




Question Number 32709 by mondodotto@gmail.com last updated on 31/Mar/18
Answered by MJS last updated on 31/Mar/18
log_2  3=((ln 3)/(ln 2))=p  log_3  2=((ln 2)/(ln 3))=p^(−1)   2^p^x  =3^p^(−x)    p^x ln 2=p^(−x) ln 3  p^(2x) =p  p^(2x−1) =1  (2x−1)ln p=0  2x−1=0  x=(1/2)
$$\mathrm{log}_{\mathrm{2}} \:\mathrm{3}=\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{ln}\:\mathrm{2}}=\mathrm{p} \\ $$$$\mathrm{log}_{\mathrm{3}} \:\mathrm{2}=\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{ln}\:\mathrm{3}}=\mathrm{p}^{−\mathrm{1}} \\ $$$$\mathrm{2}^{{p}^{{x}} } =\mathrm{3}^{{p}^{−{x}} } \\ $$$${p}^{{x}} \mathrm{ln}\:\mathrm{2}={p}^{−{x}} \mathrm{ln}\:\mathrm{3} \\ $$$${p}^{\mathrm{2}{x}} ={p} \\ $$$${p}^{\mathrm{2}{x}−\mathrm{1}} =\mathrm{1} \\ $$$$\left(\mathrm{2}{x}−\mathrm{1}\right)\mathrm{ln}\:{p}=\mathrm{0} \\ $$$$\mathrm{2}{x}−\mathrm{1}=\mathrm{0} \\ $$$${x}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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