Je dois trouver la valeur du x dans lexpression log : log2(x-3)=2-log2x

Bonjour,

[tex]\log2(x-3)=2-\log2x\\Conditions:\left\{\begin{matrix}2(x-3)>0\\2x>0 \end{matrix}\right.\ \ \ \ \left\{\begin{matrix}x-3>0\\x>0 \end{matrix}\right.\ \ \ \ \left\{\begin{matrix}x>3\\x>0 \end{matrix}\right.\ \ \ \Longrightarrow \boxed{x>3}\\\\\log2(x-3)=\log100-\log2x\\\\log2(x-3)=\log\dfrac{100}{2x}\\\\2(x-3)=\dfrac{100}{2x}\\\\2(x-3)=\dfrac{50}{x}\\\\x-3=\dfrac{25}{x}[/tex]

[tex]x(x-3)=25\\x^2-3x=25\\x^2-3x-25=0\\\Delta=(-3)^2-4\times1\times(-25)=9+100=109\\\\x_1=\dfrac{3-\sqrt{109}}{2}\approx-3,7\ (\grave{a}\ rejeter\ car\ conditions\ initiales:x>3)}\\\\x_2=\dfrac{3+\sqrt{109}}{2}\approx6,7[/tex]

Par conséquent,

[tex]\boxed{x=\dfrac{3+\sqrt{109}}{2}\approx6,7}[/tex]