Question

Log32 x -2log3 x ≤ 3.log8x + log√2x = 14.решите, пожалуйста

Answer 1

Рівняння  

$\frac{1}{3}$ log2  x+2 log2  x =14
 log2  x * ( $\frac{1}{3}$ +2)=14
$\frac{7}{3}$ log2   x=14  
Помножимо на обернений дріб $\frac{3}{7}$

log2  x=6
x=$2^{6}$
x=64

Answer 2

Problem Analysis

To solve the equation log32(x) - 2log3(x) ≤ 3log8(x) + log√2(x) = 14, we need to simplify the equation and find the value(s) of x that satisfy the inequality.

Simplifying the Equation

Let's simplify the equation step by step:

1. Convert the logarithms to a common base: - log32(x) can be written as log(x) / log(32) - log3(x) can be written as log(x) / log(3) - log8(x) can be written as log(x) / log(8) - log√2(x) can be written as log(x) / log(√2)

2. Substitute the above expressions into the equation: - log(x) / log(32) - 2(log(x) / log(3)) ≤ 3(log(x) / log(8)) + log(x) / log(√2) = 14

3. Simplify the equation further: - (log(x) - 2log(x)) / log(3) ≤ (3log(x) + log(x)) / log(8) + log(x) / log(√2) = 14 - -log(x) / log(3) ≤ 4log(x) / log(8) + log(x) / log(√2) = 14

4. Multiply both sides of the equation by log(3) to eliminate the denominators: - -log(x) ≤ 4log(x) * (log(3) / log(8)) + log(x) * (log(3) / log(√2)) = 14 * log(3)

5. Simplify the equation further: - -log(x) ≤ 4log(x) * log3(8) + log(x) * log3(√2) = 14 * log(3) - -log(x) ≤ 4log(x) * 3 + log(x) * (1/2) = 14 * log(3) - -log(x) ≤ 12log(x) + log(x) / 2 = 14 * log(3) - -log(x) ≤ (25/2)log(x) = 14 * log(3)

6. Multiply both sides of the equation by -2 to eliminate the negative sign: - log(x) ≥ (25/2)log(x) = -28 * log(3)

7. Subtract (25/2)log(x) from both sides of the equation: - log(x) - (25/2)log(x) ≥ -28 * log(3) - (25/2)log(x) - (2/2)log(x) - (25/2)log(x) ≥ -28 * log(3) - (25/2)log(x) - (-23/2)log(x) ≥ -28 * log(3) - (25/2)log(x)

8. Simplify the equation further: - (-23/2)log(x) ≥ -28 * log(3) - (25/2)log(x) - (-23/2)log(x) + (25/2)log(x) ≥ -28 * log(3) - (2/2)log(x) - (23/2)log(x) ≥ -28 * log(3) - (-21/2)log(x) ≥ -28 * log(3)

9. Divide both sides of the equation by (-21/2): - log(x) ≤ (-28 * log(3)) / (-21/2) - log(x) ≤ (56/21) * log(3) - log(x) ≤ (8/3) * log(3)

10. Convert the logarithm back to exponential form: - x ≤ 3^(8/3)

Solution

The solution to the equation log32(x) - 2log3(x) ≤ 3log8(x) + log√2(x) = 14 is x ≤ 3^(8/3).

Please note that this is a simplified form of the equation, and the solution may need to be further verified or refined.

Let me know if you need any further assistance!