Răspuns:
Explicație pas cu pas:
I)
1) 10ˣ = 100 <=> 10ˣ = 10² => x = 2
3) 64ˣ = 32 <=> (2⁶)ˣ = 2⁵ <=> 2⁶ˣ = 2⁵ => 6x = 5 => x = 5/6
4) 3ˣ = ⁴√27 <=> 3ˣ = 27¹⁾⁴ <=> 3ˣ = (3³)¹⁾⁴ <=> 3ˣ = 3³⁾⁴ => x = 3/4
8)√5ˣ = ∛625 <=> 5ˣ⁾² = ∛(5³) <=> 5ˣ⁾² = 5 => x/2 = 5 => x = 10
II)
1) 16ˣ = 64 <=> (2⁴)ˣ = 2⁶ <=> 2⁴ˣ = 2⁶ => 4x = 6 => x = 6/4 => x = 3/2
2) 32ˣ = 8 <=> (2⁵)ˣ = 2³ <=> 2⁵ˣ = 2³ => 5x = 3 => x = 3/5
5) 16⁻ˣ -64 = 0 <=> 16⁻ˣ = 64 <=> (2⁴)⁻ˣ = 2⁶ <=> 2⁻⁴ˣ = 2⁶ =>
-4x = 6 => x = -6/4 => x = -3/2
7) 3⁹ˣ - 9²⁰⁰⁷ = 0 <=> 3⁹ˣ = (3²)²⁰⁰⁷ <=> 3⁹ˣ = 3⁴⁰¹⁴ => 9x = 4014 =>
x = 4014/9 => x = 446
III)
1) 3⁻²ˣ⁻¹ = 3ˣ² <=> -2x-1 = x² <=> x²+2x+1 = 0 <=> (x+1)² = 0 => x₁=x₂ = -1
3) 16ˣ = 32ˣ <=> (2⁴)ˣ = (2⁵)ˣ => 4x = 5x <=> 5x-4x = 0 => x = 0
4) 10ˣ⁺² = 1000³ˣ <=> 10ˣ⁺² = (10³)³ˣ <=> 10ˣ⁺² = 10⁹ˣ => x+2 = 9x =>
8x = 2 => x = 2/8 => x = 1/4
5) (3/11)³ˣ⁻¹⁰ = (11/3)⁷ˣ⁻¹⁰ <=> (3/11)³ˣ⁻¹⁰ = (3/11)¹⁰⁻⁷ˣ =>
3x-10 = 10-7x <=> 3x +7x = 10+10 => 10x = 20 => x = 2
IV)
2) 5ˣ+5ˣ⁺¹ = 3750 <=> 5ˣ+5·5ˣ = 3750 <=> 5ˣ·(1+5) = 3750<=>
6·5ˣ = 3750 I:6 => 5ˣ = 625 <=> 5ˣ = 5⁴ => x =4
3) 2ˣ⁺³ + 2ˣ⁺² + 2ˣ⁺¹ + 2ˣ = 30 <=>
2³·2ˣ+2²·2ˣ+2·2ˣ+2ˣ = 30 n<=> 2ˣ·(8+4+2+1) = 30 <=>
2ˣ·15 = 30 <=> 2ˣ = 30:15 => 2ˣ = 2 => x = 1
7) 2²ˣ - 2·2ˣ - 8 = 0 ; Notam 2ˣ = t => t²-2t-8 = 0 =>
t₁,₂ = [2±√(4+32)]/2 = (2±6)/2 => t₁ = -2 ; t₂ = 4
t₁ = -2 => 2ˣ = -2 , ecuatia nu are solutii deoarece 2ˣ>0
t₂ = 4 => 2ˣ = 4 <=> 2ˣ = 2² => x = 2
13) 2·25ˣ = 10ˣ + 4ˣ <=> 2·5²ˣ = (2·5)ˣ + 2²ˣ I : 5²ˣ =>
2 = 2ˣ / 5ˣ + 2²ˣ/5²ˣ <=> (2/5)²ˣ + (2/5)ˣ -2 = 0 ;
notam (2/5)ˣ = t => t² + t -2 = 0 => t₁,₂ = [-1±√(1+8)]/2 =>
t₁,₂ = (-1±3)/2 => t₁ = -2 ; t₂ = 1
pentru t₁ nu exista solutii
t₂ = 1 => (2/5)ˣ = 1 <=> (2/5)ˣ = (2/5)⁰ => x = 0
