Question

Funcții exponențiale si logaritmice
In loc de N pui 9

Answer 1

Răspuns:

1. log₇(7x+9)=2

DDVA

7x+9>0

7x>-9

x>-9/7

x=(-9/7,+infinit)=dva

rezolvi ecuatia

1.  7x+9=7²

7x=49-9

7x=40

x=40/7> -9/7 apartine  DVA

2.  log₀₅(x-2)>log₀₅9/(x-3)

dva

x-2>0    x>2

9/(x-3)>0

x-3>0   x>3

DVA=(3,+infinit)

rezolvi inecuatia

x-2<9/(x-3)

(x-2)(x-3)<9

x²-2x-3x+6<9

x²-5x-3<0

x²-5x-3=0

x1==(5-$\sqrt{37}$)/2<3 nu apartine   DVA

x2=(5+√37)/2>3 solutie

x apartine (5+rad 37/2,+oo)

3. log₍ₓ₊₂₎lx-9l≥1

DVA

x+2>0   x> -2

x+2≠1      x≠-1

x-9≠0 x≠9

DVA=(-2,1)U(1,9)I(9,∞)

rezolvare

x∈(-2,-1)

x+2∈(0,1)

l x-9l≤1

-1≤x-9≤1

x∈(8,10)∩(--2 ,-1)=Ф Nu admite  solutii

Caz 2

x∈(-1,1)U(1.9)U(9,+∞)

x+2>1

x-9≥x+2 imposibil

inecuatia nu admite  solutii

ex 4

log₁/₂(3x+5)≥ -2

DVA

3x+5>0

3x> -5

x>-5/3

x∈(-5/3,+∞)

rezolvare

3x+5)≤1/2⁻²

3x+5≤1/4

3x≤1/4-5

3x≤-19/20

intersectezi cu dva

(-∞. -19/20)∩(-5/3+∞)=

(-5/3, -19/20)

7

DVA x∈R

(2√2-√7)(2√2+√7)=8-7=1

(2√2+√7)=1/(2√2-√7)

(2√2+√7)ˣ=y>0

(2√2-√7)ˣ=1/y

Ecuatia   devine

y+1/y=4√2

y²-4√2y+1=0

Δ=4√2)²-4=28

y1=(4-√28)2=(4-2√7)/2=2-√7<0 nu se accepta

y2=2+√7.>0 solutie=>

(2√2+√7)ˣ=(2+√7

x=log(2+√7)

Explicație pas cu pas: