Question

2log2 (log2 x) + log0,5 ( log2 x) = 1

Answer 1

Ответ:

x=4

Объяснение:

2log2 (log2 x) + log0,5 ( log2 x) = 1

x E ( 1, +бесконечности)

2log2(log2x) -log2(log2x)=1

log2(log2x)=1

log2x=2

x=2^2

x=4

Answer 2

To solve the equation 2log2(log2(x)) + log0.5(log2(x)) = 1, we'll use some logarithmic properties and algebraic manipulation. First, let's simplify the equation:

1. Rewrite log0.5 as log2(0.5): 2log2(log2(x)) + log2(0.5 * log2(x)) = 1

2. Use the properties of logarithms to simplify further: log2(2^(2log2(log2(x))) * 0.5 * log2(x)) = 1

3. Combine the logarithmic expressions inside the logarithm: log2(2^(2log2(log2(x)) - 1) * log2(x)) = 1

4. Use the fact that log2(a^b) = b * log2(a): 2log2(log2(x)) - 1 + log2(x) = 1

5. Move the constant term to the right side of the equation: 2log2(log2(x)) + log2(x) = 1 + 1

6. Combine the constants on the right side: 2log2(log2(x)) + log2(x) = 2

7. Now, we can use a substitution to simplify further. Let u = log2(x):

2log2(u) + u = 2

8. This is a quadratic equation in terms of u. Rearrange it:

2log2(u) + u - 2 = 0

9. We can solve this quadratic equation. Let's use substitution again to simplify it further. Let v = log2(u):

2v + v - 2 = 0

10. Combine like terms: 3v - 2 = 0

11. Add 2 to both sides: 3v = 2

12. Divide by 3: v = 2/3

13. Now, we need to find u: v = log2(u)

2/3 = log2(u)

14. Convert to exponential form: u = 2^(2/3)

15. Finally, find x by reversing the initial substitution: x = 2^u = 2^(2/3)

So, the solution to the equation is x = 2^(2/3).