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Let's go through and solve each of the problems in the image you've uploaded:

### 3. Rezolvați în mulțimea numerelor reale ecuația:
\[
\log_{4}(x+3) + \log_{4}(x-3) = 2.
\]

#### Solution:
We start by using the logarithm property \(\log_a b + \log_a c = \log_a(b \cdot c)\):
\[
\log_{4}((x+3)(x-3)) = 2.
\]
This simplifies the equation to:
\[
\log_{4}(x^2 - 9) = 2.
\]
To solve for \(x\), convert the logarithmic equation to its exponential form:
\[
x^2 - 9 = 4^2 = 16.
\]
Now, solve for \(x\):
\[
x^2 = 25 \implies x = \pm 5.
\]
Check the solutions:
- For \(x = 5\): \(x+3 = 8\) and \(x-3 = 2\), both are positive, so it's valid.
- For \(x = -5\): \(x+3 = -2\), which is negative, making the logarithm undefined.

Thus, the solution is \(x = 5\).

### 4. Calculați probabilitatea ca, alegând un număr din mulțimea numerelor naturale de două cifre, acesta să aibă produsul cifrelor egal cu 6.
  
#### Solution:
Consider the two-digit numbers \(ab\) where \(a\) and \(b\) are digits and \(a \neq 0\). We want \(a \cdot b = 6\).

Possible pairs \((a, b)\) are:
\[
(1, 6), (2, 3), (3, 2), (6, 1).
\]
These give the numbers: 16, 23, 32, 61.

The total number of two-digit numbers is \(90\) (from \(10\) to \(99\)).

The probability is:
\[
\frac{4}{90} = \frac{2}{45}.
\]

### 5. Determinați numărul real \(a\), pentru care vectorii \(\vec{u} = a\hat{i} + 2\hat{j}\) și \(\vec{v} = 3\hat{i} - 3\hat{j}\) sunt coliniari.

#### Solution:
For the vectors to be collinear, \(\vec{u}\) must be a scalar multiple of \(\vec{v}\):
\[
\vec{u} = k \vec{v}.
\]
This gives:
\[
a\hat{i} + 2\hat{j} = k(3\hat{i} - 3\hat{j}).
\]
Equating components:
\[
a = 3k, \quad 2 = -3k.
\]
Solving for \(k\):
\[
k = -\frac{2}{3}.
\]
Then:
\[
a = 3 \times -\frac{2}{3} = -2.
\]
So, \(a = -2\).

### 6. Arătați că \((\sin x - \cos x)^2 + \sin 2x = 1\), pentru orice număr real \(x\).

#### Solution:
Expand the square:
\[
(\sin x - \cos x)^2 = \sin^2 x - 2\sin x \cos x + \cos^2 x.
\]
Using the identity \(\sin^2 x + \cos^2 x = 1\) and \(\sin 2x = 2\sin x \cos x\):
\[
(\sin x - \cos x)^2 + \sin 2x = 1 - 2\sin x \cos x + 2\sin x \cos x = 1.
\]
The equation is indeed true for any real \(x\).

### Questions:
1. How would you solve similar logarithmic equations with different bases?
2. Can the method for solving the probability problem be applied to other products of digits?
3. What other conditions must be met for two vectors to be collinear?
4. How can trigonometric identities help simplify complex expressions?
5. What are the key properties of logarithms that simplify equations?

### Tip:
Always verify the domain of solutions when dealing with logarithms to ensure the arguments are positive.

Learn how to solve logarithmic equations, calculate probabilities for two-digit numbers, determine vector collinearity, and simplify trigonometric identities with detailed step-by-step solutions. Suitable for high school students.

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