Question

Log2(x-5)-log2(x+5)+log2(x^2-25)=0

Answer 1

Применены : определение и свойства логарифмов

Answer 2

To solve this logarithmic equation, we can use the properties of logarithms to simplify the equation and then solve for x.

First, we can combine the first two logarithmic terms using the quotient rule:

log2((x-5)/(x+5)) + log2(x^2-25) = 0

Then, we can use the product rule to combine the two logarithmic terms on the left-hand side:

log2(((x-5)/(x+5)) * (x^2-25)) = 0

Now we can rewrite the equation in exponential form:

2^0 = ((x-5)/(x+5)) * (x^2-25)

Simplifying the left-hand side:

1 = ((x-5)/(x+5)) * (x^2-25)

Multiplying both sides by (x+5):

x+5 = (x-5) * (x^2-25)

Expanding the right-hand side:

x+5 = x^3 - 30x - 25x + 125

Combining like terms:

x^3 - 56x + 120 = 0

We can use the rational root theorem or synthetic division to find that x=4 is a root of the equation. Dividing by (x-4) using synthetic division gives:

x^2 + 4x - 30 = 0

Using the quadratic formula, we find the other two roots to be:

x = (-4 ± sqrt(16 + 120)) / 2

x = (-4 ± sqrt(136)) / 2

x ≈ -9.8 or x ≈ 5.8

Therefore, the solutions to the equation are x = 4, x ≈ -9.8, and x ≈ 5.8.