a) [tex]\displaystyle 5^{x^2+4x}=1[/tex] [tex]\displaystyle 5^{x^2+4x}=5^0[/tex] Если равны основания степеней, то равны и сами степени [tex]x^2+4x=0 < = > x(x+4)=0 < = > \left[\begin{array}{ccc}x=0\\x+4=0\\\end{array}\right. < = > \left[\begin{array}{ccc}x=0\\x=-4\\\end{array}\right.[/tex]
Ответ: х = -4; х = 0
b) [tex]\displaystyle log_{\frac{1}{2} }(3x-2)=-2[/tex] [tex]\displaystyle 3x-2=(\frac{1}{2} )^{-2}[/tex] [tex]\displaystyle 3x-2=4[/tex] [tex]\displaystyle 3x=6|:3[/tex] [tex]\displaystyle x=2[/tex] Ответ: х = 2
c) [tex]\displaystyle sin(x)=\frac{1}{\sqrt{2} }[/tex] [tex]\displaystyle x=(-1)^narcsin(\frac{\sqrt{2} }{2} )+\pi n[/tex], n∈Z [tex]\displaystyle x=\left[\begin{array}{ccc}\frac{\pi }{4}+2\pi n \\\frac{3\pi }{4}+2\pi n\\\end{array}\right[/tex], n∈Z
Answers & Comments
a) [tex]\displaystyle 5^{x^2+4x}=1[/tex]
[tex]\displaystyle 5^{x^2+4x}=5^0[/tex]
Если равны основания степеней, то равны и сами степени
[tex]x^2+4x=0 < = > x(x+4)=0 < = > \left[\begin{array}{ccc}x=0\\x+4=0\\\end{array}\right. < = > \left[\begin{array}{ccc}x=0\\x=-4\\\end{array}\right.[/tex]
Ответ: х = -4; х = 0
b) [tex]\displaystyle log_{\frac{1}{2} }(3x-2)=-2[/tex]
[tex]\displaystyle 3x-2=(\frac{1}{2} )^{-2}[/tex]
[tex]\displaystyle 3x-2=4[/tex]
[tex]\displaystyle 3x=6|:3[/tex]
[tex]\displaystyle x=2[/tex]
Ответ: х = 2
c) [tex]\displaystyle sin(x)=\frac{1}{\sqrt{2} }[/tex]
[tex]\displaystyle x=(-1)^narcsin(\frac{\sqrt{2} }{2} )+\pi n[/tex], n∈Z
[tex]\displaystyle x=\left[\begin{array}{ccc}\frac{\pi }{4}+2\pi n \\\frac{3\pi }{4}+2\pi n\\\end{array}\right[/tex], n∈Z
Ответ: х = π/4+2πn;3π/4+2πn, n∈Z