Question

Log2 (2^x +3) × log2 (2^x+2 +12)=8

Answer 1

Дано выражение log2 ((2^x)+3) × log2 (2^(х+2)+12)=8
Преобразуем log2 (2^(х+2)+12) =  log2 (4*(2^х)+3) =
= log2(4)+log2(2^x+3)
Заменим 2^x = а, log2(а+3) = в

Получаем квадратное уравнение: в² + 2в - 8 = 0.
Квадратное уравнение, решаем относительно в: 
Ищем дискриминант:D=2^2-4*1*(-8)=4-4*(-8)=4-(-4*8)=4-(-32)=4+32=36;
Дискриминант больше 0, уравнение имеет 2 корня:
в_1=(√36-2)/(2*1)=(6-2)/2=4/2=2;
в_2=(-√36-2)/(2*1)=(-6-2)/2=-8/2=-4 это значение отбрасываем, так как 2^(-4) - 3 даёт отрицательное логарифмируемое выражение, что невозможно.

Делаем обратную замену log2(2^x+3) = 2.
2² =  2^x + 3
2^x = 2 ² - 3 = 1 = 2 ⁰.
Отсюда ответ: х = 0.

Answer 2

To solve the equation log2(2^x + 3) × log2(2^x+2 + 12) = 8, we can simplify it step by step. Here's the solution process:

Step 1: Apply the properties of logarithms to simplify the equation. Using the product rule of logarithms, we can rewrite the equation as follows:

log2(2^x + 3) + log2(2^x+2 + 12) = 8

Step 2: Apply the power rule of logarithms to simplify further. Using the power rule, we can rewrite the equation as:

log2(2^x + 3) + log2((2^x+2) × 12) = 8

Step 3: Combine the logarithms into a single logarithm using the addition rule. Using the addition rule of logarithms, we can rewrite the equation as:

log2[(2^x + 3) × ((2^x+2) × 12)] = 8

Step 4: Simplify the expression inside the logarithm. Applying the distributive property and simplifying, we get:

log2(24 × 2^x × 2^x + 36 × 2^x + 36) = 8

Simplifying further, we have:

log2(24 × 2^(2x) + 36 × 2^x + 36) = 8

Step 5: Apply the logarithmic identity to eliminate the logarithm. Using the logarithmic identity logb(b^a) = a, we can rewrite the equation as:

24 × 2^(2x) + 36 × 2^x + 36 = 2^8

Simplifying the right side:

24 × 2^(2x) + 36 × 2^x + 36 = 256

Step 6: Rearrange the equation to form a quadratic equation. Subtracting 256 from both sides, we have:

24 × 2^(2x) + 36 × 2^x - 220 = 0

Step 7: Substitute a new variable to simplify the equation. Let's substitute y = 2^x:

24y^2 + 36y - 220 = 0

Step 8: Solve the quadratic equation. We can factorize the quadratic equation:

(4y - 10)(6y + 22) = 0

Setting each factor to zero:

4y - 10 = 0 -> 4y = 10 -> y = 10/4 -> y = 2.5

6y + 22 = 0 -> 6y = -22 -> y = -22/6 -> y = -11/3 ≈ -3.67

Step 9: Solve for x. Substituting back y = 2^x:

For y = 2.5: 2^x = 2.5 x = log2(2.5)

For y = -11/3: 2^x = -11/3 (Not a valid solution since 2^x cannot be negative)

Thus, the solution is x = log2(2.5) (approximately).