Question

Log2(2x+3)+log2(x-1)=4-log2 16/3

Answer 1

Ответ:

log2(2x+3)+log2(x-1)=4-log2(16)/3

log(x-1)/log(2) + log(2x+3)/log(2) = 8/3

x = 1/4(√(25+32x2^2/3) - 1)

Answer 2

To solve the equation:

log₂(2x + 3) + log₂(x - 1) = 4 - log₂(16/3)

We can use logarithm properties to simplify and solve it. First, let's manipulate the equation step by step:

1. Combine the logarithms on the left side using the product rule for logarithms:

log₂((2x + 3)(x - 1)) = 4 - log₂(16/3)

2. Move the right side logarithm to the left side by adding it to both sides:

log₂((2x + 3)(x - 1)) + log₂(16/3) = 4

3. Now, use the properties of logarithms to combine the two logarithms on the left side into a single logarithm:

log₂(((2x + 3)(x - 1) * 16/3)) = 4

4. Remove the logarithm by exponentiating both sides with base 2:

((2x + 3)(x - 1) * 16/3) = 2^4

5. Simplify the right side:

(2x + 3)(x - 1) * 16/3 = 16

6. Multiply both sides by 3/16 to isolate the left side:

(2x + 3)(x - 1) = 3

7. Expand and simplify the left side:

2x² - 2x + 3x - 3 = 3

8. Combine like terms:

2x² + x - 6 = 3

9. Move 3 to the right side:

2x² + x - 6 - 3 = 0

10. Simplify further:

2x² + x - 9 = 0

Now, you can attempt to solve this quadratic equation for x. You can either use the quadratic formula or factor the equation if possible. In this case, factoring is not straightforward, so let's use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Where a = 2, b = 1, and c = -9:

x = (-1 ± √(1² - 4 * 2 * (-9))) / (2 * 2)

x = (-1 ± √(1 + 72)) / 4

x = (-1 ± √73) / 4

So, the solutions for x are:

x₁ = (-1 + √73) / 4 x₂ = (-1 - √73) / 4

These are the exact solutions for the given equation. If you need decimal approximations, you can calculate them using a calculator.