Question

Log2 (x^2 - x - 4) <3

Answer 1

Ответ:

(4;1- √17/2)

Пошаговое объяснение:

log2(x^2-x-4)<3

Есть такая теорема хорошая ,которая гласит

(loga x1,a≠0, x1>0,x>0)

Начнём решать как в теореме,основание>1,значит функция возрастает, значит знак неравенства такой же останется

x^2-x-4>0

Решаем методом змейки

1)Приравняем к нулю

x^2-x-4=0

2)Разложим многочлен и  получим

х1,2=1+-√(-1)^2-4*1*(-4)/2

х1=1+√17/2

х2=1- √17/2

Разложим по формуле

ax^2+bx+c=a(x-x1)·(x-x2)

Получим

x^2-x-4=(x-1+ √17/2)

Тогда

x-1+ √17/2>0

x=1- √17/2

тогда x€(1- √17/2;+∞)

приступим решать само уравнение

log2(x^2-x-4)

Потенцируем и получим

x^2-x-4<8

x^2-x-12=0

x1,2=1+- √(-1)^2-4·1·(-12)/2

x1,2=1+- √49/2

x1=4

x2=-3(не входит)

примерно равно (-1,56:+∞)

Разложим многочлен

(х-4)<0

х=4

Отметив на координатной оси точку x=4 определим корни

x€(4;1- √17/2)

Ответ: (4;1- √17/2)

Answer 2

To solve the inequality Log2(x^2 - x - 4) < 3, we need to apply the properties of logarithms. Remember that Log2 represents the logarithm base 2.

Step 1: Write the inequality in exponential form: Log2(x^2 - x - 4) < 3 can be written as: 2^(Log2(x^2 - x - 4)) < 2^3

Step 2: Simplify the exponential expression on the right side: 2^(Log2(x^2 - x - 4)) < 8

Step 3: Remove the logarithm on the left side using the property that 2^(Log2(a)) = a: x^2 - x - 4 < 8

Step 4: Move all terms to one side of the inequality to set it to zero: x^2 - x - 4 - 8 < 0

Step 5: Combine like terms: x^2 - x - 12 < 0

Step 6: Factor the quadratic expression on the left side (or use the quadratic formula) and solve for x:

x^2 - x - 12 = 0 (x - 4)(x + 3) = 0

Setting each factor to zero and solving for x gives us two possible values: x - 4 = 0 ---> x = 4 x + 3 = 0 ---> x = -3

Step 7: Create a sign chart or use test points to determine the solution intervals:

Using test points: Choose a value less than -3, say x = -4: (-4)^2 - (-4) - 4 = 16 + 4 - 4 = 16, which is positive.

Choose a value between -3 and 4, say x = 0: 0^2 - 0 - 4 = -4, which is negative.

Choose a value greater than 4, say x = 5: 5^2 - 5 - 4 = 25 - 5 - 4 = 16, which is positive.

Step 8: Analyze the sign chart or test points to find the solution to the inequality:

The inequality x^2 - x - 12 < 0 is true for values of x between -3 and 4, excluding the endpoints -3 and 4.

Thus, the solution to the original inequality Log2(x^2 - x - 4) < 3 is the set of x values between -3 and 4, excluding -3 and 4. In interval notation, this is (-3, 4).