Question

Sa se rezolve in R ecuatia: va rog mult, cine ma poate ajuta?

$log \frac{1}{2} x^{2} -lg4=log \frac{1}{2}x+lg25$
$log ^{2} _{3}(x+ 2)-3log _{3} (x+2)-4=0$
$12-lg ^{2}x=lgx$

Answer 1

1) CE:    x²>0,x>0
 $log_{ \frac{1}{2} }x^{2}-log_{ \frac{1}{2} } x=lg4+lg25 \\ log_{ \frac{1}{2} } x=lg100 \\ log_{ \frac{1}{2} } x=2$
$x=( \frac{1}{2})^{2} = \frac{1}{4}$             

conditiile de existenta sunt indeplinite,deci  Sol: {1/4}
2)CE:  x+2>0=> x>-2.

notez log₃(x+2)=t.
ecuatia devine  t²-3t-4=0
  Δ=9+16=25

t₁=4
t₂=-1.

pt t=4          log₃(x+2)=4;      3⁴=x+2;        x+2=81 => x=79.
pt t=-1         log₃(x+2)=-1;      3⁻¹=x+2;    x+2=1/3 => x=-5/3

conditiile sunt indeplinite,deci  Sol: {-5/3 ; 79}

3) CE:  x>0.
notez lg x=t
12-t²=t
t²+t-12=0.
Δ=1+48=49.

t₁=4
t₂=-3.

pt t=4      lg x=4;        x=10⁴=10000.
pt t=-3      lg x=-3;      x=10⁻³=1/1000
 
conditiile sunt indeplinite ,deci Sol: {1/1000; 10000}