Ответ:
х = 8
Объяснение:
Теория: [tex]\displaystyle a^{log_a(b)} = b[/tex][tex]\displaystyle a*log_b(c) = log_b(c^a)[/tex]
Решение:[tex]\displaystyle \left \{ {{x^{2*log_4(x)}=x*16^{log_4(x)}} \atop {x > 0}} \right. < = > \left \{ {{x^{log_4(x^2)}=x*4^{2log_4(x)}} \atop {x > 0}} \right. < = > \left \{ {{x^{log_4(x^2)}=x*4^{log_4(x^2)}} \atop {x > 0}} \right. < = >[/tex][tex]\displaystyle < = > \left \{ {{x^{log_4(x^2)}=x*x^2} \atop {x > 0}} \right. < = > \left \{ {{x^{log_4(x^2)}=x^3} \atop {x > 0}} \right.[/tex]
Т. к. равны основания степеней ⇒ равны и сами степени[tex]\displaystyle \left \{ {{log_4(x^2)=3} \atop {x > 0}} \right. < = > \left \{ {{x^2=4^3} \atop {x > 0}} \right. < = > \left \{ {{x^2=64} \atop {x > 0}} \right. < = > x=8[/tex]
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Answers & Comments
Ответ:
х = 8
Объяснение:
Теория:
[tex]\displaystyle a^{log_a(b)} = b[/tex]
[tex]\displaystyle a*log_b(c) = log_b(c^a)[/tex]
Решение:
[tex]\displaystyle \left \{ {{x^{2*log_4(x)}=x*16^{log_4(x)}} \atop {x > 0}} \right. < = > \left \{ {{x^{log_4(x^2)}=x*4^{2log_4(x)}} \atop {x > 0}} \right. < = > \left \{ {{x^{log_4(x^2)}=x*4^{log_4(x^2)}} \atop {x > 0}} \right. < = >[/tex]
[tex]\displaystyle < = > \left \{ {{x^{log_4(x^2)}=x*x^2} \atop {x > 0}} \right. < = > \left \{ {{x^{log_4(x^2)}=x^3} \atop {x > 0}} \right.[/tex]
Т. к. равны основания степеней ⇒ равны и сами степени
[tex]\displaystyle \left \{ {{log_4(x^2)=3} \atop {x > 0}} \right. < = > \left \{ {{x^2=4^3} \atop {x > 0}} \right. < = > \left \{ {{x^2=64} \atop {x > 0}} \right. < = > x=8[/tex]