Question

Lg(x-5)-lg(5x-19)=0 log3(x-2)+log3(x+16)=2 lg(x+2)=lg(x-6)

Answer 1

Ответ:

Пошаговое объяснение:

lg(x-5)-lg(5x-19)=0

ОДЗ: x-5>0  x>5   5x-19>0   5x>19    x>3,8   ⇒  x∈(5;+∞)

lg(x-5)=lg(5x-19)

x-5=5x-19

4x=14  |÷4

x=3,5 ∉ОДЗ

Ответ: нет решения.

log₃(x-2)+log₃(x+16)=2

ОДЗ: x-2>0    x>2     x+16>0    x>-16    ⇒    x∈(2;+∞)

log₃((x-2)(x+16))=2

x²+14x-32=3²

x²+14x-32=9

x²+14x-41=0   D=32

x₁=-7-2√2 ∉ОДЗ       х₂=-7+2√2  ∉ОДЗ

Ответ: нет решения.

lg(x+2)=lg(x-6)

x+2=x-6

2≠-6

Ответ: нет решения.

Answer 2

Let's solve the given equations step by step:

1. Equation: lg(x - 5) - lg(5x - 19) = 0

We can simplify this equation using logarithmic properties. The subtraction of logarithms is equivalent to the division of the arguments:

lg((x - 5) / (5x - 19)) = 0

Since lg(1) = 0, we can rewrite the equation as:

(x - 5) / (5x - 19) = 1

Now we can solve for x:

(x - 5) = (5x - 19)

Simplifying the equation:

x - 5 = 5x - 19

Rearranging terms:

4x = 14

Dividing by 4:

x = 14 / 4

x = 7/2 or x = 3.5

Therefore, one solution is x = 3.5.

2. Equation: log3(x - 2) + log3(x + 16) = 2

Using the logarithmic property, the sum of logarithms is equivalent to the multiplication of the arguments:

log3((x - 2) * (x + 16)) = 2

We can rewrite the equation using the exponential form:

3^2 = (x - 2) * (x + 16)

9 = x^2 + 14x - 32

Rearranging terms:

x^2 + 14x - 41 = 0

Unfortunately, this equation doesn't factor easily, so we'll use the quadratic formula to solve for x:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = 14, and c = -41:

x = (-14 ± √(14^2 - 4(1)(-41))) / (2(1))

x = (-14 ± √(196 + 164)) / 2

x = (-14 ± √(360)) / 2

x = (-14 ± 6√10) / 2

x = -7 ± 3√10

Therefore, the solutions are x ≈ -7 + 3√10 and x ≈ -7 - 3√10.

3. Equation: lg(x + 2) = lg(x - 6)

Since both sides have the same logarithm base and are equal, we can set their arguments equal to each other:

x + 2 = x - 6

By subtracting x from both sides, we get:

2 = -6

This equation is inconsistent and has no solution.

To summarize:

• Equation 1 has one solution: x = 3.5.
• Equation 2 has two solutions: x ≈ -7 + 3√10 and x ≈ -7 - 3√10.
• Equation 3 has no solution.