Question

log X + log (X+3) = 2 log (5-X)

2logX = 3+log X/10

log (2X- 7) – log(X-1) = log5

log X + log (X-3) = 2log X +1

log 16- X² / log (3X – 4 )² = 2

log 2 + log (11 – X²) = 2 log (5-X)

log X + log (X+9) = 1

log (5X+6) / log X = 2

log 8 + log x + log X² = 3

log 4 + log X = 2

AYUDA ES PARA EL MIERCOLEEEES

Answer 1

Al resolver las expresiones logarítmicas se obtiene:

1.  x = 25/13 ⇒ No tiene solución real

2. x = 100

3. x = -2/3 ⇒ No tiene solución real

4. x = -1/5

5. x ≈ 0.749 y x ≈ 1.865

6. x = 3  y x = 1/3

7. x = 1 y x = -10

8. x = 6 y x = -1

9. x = 5

10. x = 25

Explicación paso a paso:

1. log (x) + log (x+3) = 2 log (5-x)

Aplicar propiedades de logaritmos;

log (a) + log (b) = log (a · b)

log(a)ᵇ = b log (a)

log [(x)(x + 3)] = log (5-x)²

log (x² + 3x) = log (25 - 10x + x²)

Aplicar base 10;

₁₀[log (x² + 3x) = ₁₀[log (25 - 10x + x²)]

x² + 3x = 25 - 10x + x²

10x + 3x = 25

13x = 25

x = 25/13

2. 2 log (x) = 3+log x/10

Aplicar propiedades de logaritmos;

• log(a)ᵇ = b log (a)
• log(a/b) = log(a) - log(b)

log(x²) = 3 + log(x) - log(10)

log(x²) = 3 + log(x) - 1

log(x²) = 2 + log(x)

log(x²) - log(x)  = 2

log(x²/x) = 2

₁₀log (x) = ₁₀2

x = 100

3. log (2x- 7) – log(x-1) = log(5)

Sumar log(x-1) a ambos lados;

log(2x-7) - log(x-1) + log(x-1) = log(5) + log(x-1)

log(2x-7) = log(5) + log(x-1)

Aplicar propiedades de logaritmos;

• log (a) + log (b) = log (a · b)

log(2x-7) = log [(5)(x-1)]

₁₀log(2x-7) = ₁₀log (5x-5)

2x - 7 = 5x -5

3x = -2

x = -2/3

4. log (x) + log (x-3) = 2log (x +1)

Aplicar propiedades de logaritmos;

• log (a) + log (b) = log (a · b)
• log(a)ᵇ = b log (a)

log [x(x-3)] = log(x+1)²

₁₀log (x²-3x) = ₁₀log(x²+2x+1)

x²-3x = x²+2x+1

5x = -1

x = -1/5

5. log (16- x²) / log (3x – 4 )² = 2

Multiplicar por log (3x – 4 )² a ambos lados;

[log (3x – 4 )²] [log (16- x²)/log (3x – 4 )²] = 2 log (3x – 4 )²

log (16- x²) = 2 log (3x – 4 )²

Aplicar propiedades de logaritmos;

• log(a)ᵇ = b log (a)

₁₀log (16- x²) = ₁₀log (3x – 4 )⁴

16- x² = (3x – 4 )⁴

16- x² = x⁴-4(27x³)(4) +6(9x²)(16) -4(3x)(64) + 256

16- x² = x⁴-432x³ +864x² -768x + 256

x⁴-432x³ +865x² -768x + 240 =0

x ≈ 0.749 y x ≈ 1.865

6. log (2) + log (11 – x²) = 2 log (5-x)

Aplicar propiedades de logaritmos;

• log (a) + log (b) = log (a · b)
• log(a)ᵇ = b log (a)

₁₀log[2(11-x²)] = ₁₀log(5-x)²

22 - 2x² = 25 - 10x + x²

3x² -10x + 3 = 0

x = 3  y x = 1/3

7. log (x) + log (x+9) = 1

Aplicar propiedades de logaritmos;

• log (a) + log (b) = log (a · b)

₁₀log [x(x+9)] = ₁₀1

x² + 9x = 10

x² + 9x - 10 =0

x = 1 y x = -10

8. log (5x+6)/log (x) = 2

Multiplicar log (x) a ambos lados;

[log (5x+6)/log (x)] log(x) = 2 log (x)

log (5x+6) = 2 log (x)

Aplicar propiedades de logaritmos;

• log(a)ᵇ = b log (a)

₁₀log (5x+6) = ₁₀log (x²)

5x + 6 = x²

x² -5x - 6 =0

x = 6 y x = -1

9. log (8) + log (x) + log (x²) = 3  

Aplicar propiedades de logaritmos;

• log (a) + log (b) = log (a · b)

log[(8)(x)(x²) = 3

₁₀log(8x³) = ₁₀3

8x³ = 1000

x³ = 1000/8

x³ = 125

x = ∛125

x = 5

10. log (4) + log (x) = 2

Aplicar propiedades de logaritmos;

• log (a) + log (b) = log (a · b)

₁₀log(4x) = ₁₀2

4x = 100

x = 25