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Реши все задачи с полным решением

Let's go through the tasks one by one, providing detailed solutions.

### Task 8.1 (Решить уравнение):

#### 1) \( y \ln y - y = 0 \)

**Solution:**

1. Factor out \( y \):
   \[
   y(\ln y - 1) = 0
   \]

2. Set each factor to zero:
   - \( y = 0 \)
   - \( \ln y - 1 = 0 \)

3. Solve for \( y \) in \( \ln y - 1 = 0 \):
   \[
   \ln y = 1
   \]

4. Exponentiate both sides to solve for \( y \):
   \[
   y = e
   \]

**Final Solution:** \( y = 0 \) or \( y = e \).

#### 2) \( y \ln y = \frac{y}{2} \)

**Solution:**

1. Divide both sides by \( y \) (assuming \( y \neq 0 \)):
   \[
   \ln y = \frac{1}{2}
   \]

2. Exponentiate both sides:
   \[
   y = e^{\frac{1}{2}} = \sqrt{e}
   \]

**Final Solution:** \( y = \sqrt{e} \).

#### 3) \( 5y = y \sqrt{y} \)

**Solution:**

1. Divide both sides by \( y \) (assuming \( y \neq 0 \)):
   \[
   5 = \sqrt{y}
   \]

2. Square both sides:
   \[
   y = 25
   \]

**Final Solution:** \( y = 25 \).

#### 4) \( y \ln y = 2y - 2 \)

**Solution:**

1. Divide both sides by \( y \) (assuming \( y \neq 0 \)):
   \[
   \ln y = 2 - \frac{2}{y}
   \]

2. No simple analytical solution is possible. This is typically solved using numerical methods or approximation.

**Final Solution:** Numerical methods required.

### Task 8.2 (Решить уравнение):

#### 5) \( y \ln x = x \)

**Solution:**

1. Divide both sides by \( x \) (assuming \( x \neq 0 \)):
   \[
   \frac{y}{x} = \frac{1}{\ln x}
   \]
   - \( y = \frac{x}{\ln x} \).

**Final Solution:** \( y = \frac{x}{\ln x} \).

#### 6) \( y = \ln \left(\frac{x^2 - 1}{4x}\right) \)

**Solution:**

This is already solved for \( y \). To explore further, we could analyze the domain:
- \( x^2 - 1 > 0 \) and \( 4x > 0 \) which implies \( x > 1 \).

**Final Solution:** \( y = \ln \left(\frac{x^2 - 1}{4x}\right) \) where \( x > 1 \).

#### 7) \( y = \ln \left(x^2 - x - 6\right) - 4x \)

**Solution:**

Again, this is an explicit expression for \( y \). The domain requires \( x^2 - x - 6 > 0 \) which implies \( x > 3 \) or \( x < -2 \).

**Final Solution:** \( y = \ln \left(x^2 - x - 6\right) - 4x \) with \( x > 3 \) or \( x < -2 \).

#### 8) \( y = \log_{2}\left(\frac{1}{x^2}\right) \)

**Solution:**

Apply the logarithm properties:
\[
y = \log_{2}(1) - 2\log_{2}(x) = -2\log_{2}(x)
\]

**Final Solution:** \( y = -2\log_{2}(x) \).

#### 9) \( y = x^2 \ln x \)

**Solution:**

This is already in the form \( y = f(x) \).

**Final Solution:** \( y = x^2 \ln x \).

---

Would you like any further explanations or have any questions? Here are 5 related questions to explore:

1. How would you solve for \( y \) if the logarithmic equation involved base 10 instead of \( e \)?
2. What is the importance of determining the domain of a function when solving equations?
3. How can you solve equations involving logarithms using numerical methods?
4. What steps are required to determine if a logarithmic equation has extraneous solutions?
5. How would you graphically represent the solutions to these equations?

**Tip:** Always consider the domain of the variables when working with logarithmic equations, as the logarithm function is only defined for positive real numbers.

Explore detailed solutions for various logarithmic and algebraic equations, covering key mathematical concepts like logarithmic properties and exponentiation. Suitable for students in Grades 10-12.

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