This image contains various logarithmic equations, inequalities, and expressions. I will solve and explain them step by step for better understanding. Let's address the problems in separate sections.

1. Equation Section (Top Panel)

The top contains logarithmic equations like: $\log_{\frac{1}{4}}(x) = -2, \quad \log_{2}(x) = -3, \quad \log_{2}(-5x-3) = 6.$ I will solve these equations below:

1.1 $\log_{\frac{1}{4}}(x) = -2$:

Using the logarithmic definition: $x = \left(\frac{1}{4}\right)^{-2} = 4^2 = 16.$ So, $x = 16$.

1.2 $\log_{2}(x) = -3$:

Using the logarithmic definition: $x = 2^{-3} = \frac{1}{8}.$ So, $x = \frac{1}{8}$.

1.3 $\log_{2}(-5x - 3) = 6$:

Using the logarithmic definition: $-5x - 3 = 2^6 = 64.$ Solving for $x$: $-5x = 64 + 3 = 67 \implies x = -\frac{67}{5}.$

2. Logical Expressions (Логика Section)

This section includes: $\log_{2}\left(\log_{3}(x)\right) = 1, \quad \log(-2x - 4) = 2, \quad \ln\left(\frac{1}{e^{-3}}\right), \quad \text{and more}.$

2.1 $\log_{2}\left(\log_{3}(x)\right) = 1$:

Using logarithmic properties: $\log_{3}(x) = 2^1 = 2.$ Now solve: $x = 3^2 = 9.$

2.2 $\log(-2x - 4) = 2$:

Convert to exponential form: $-2x - 4 = 10^2 = 100.$ Solving for $x$: $-2x = 100 + 4 = 104 \implies x = -52.$

2.3 $\ln\left(\frac{1}{e^{-3}}\right)$:

Simplify: $\ln\left(\frac{1}{e^{-3}}\right) = \ln(e^3) = 3.$

3. Inequalities (Неравенства Section)

3.1 $\log_{0.5}(x) > \log_{0.5}(5)$:

In base $0.5$, the logarithmic function is decreasing, so: $x < 5.$

3.2 $\log_{7}(2x + 5) < \log_{7}(9)$:

In base $7$, the logarithmic function is increasing, so: $2x + 5 < 9 \implies 2x < 4 \implies x < 2.$

4. Quadratic Logarithmic Equations (КВ УР-я Section)

$\log_{4}^2(x) + 3\log_{2}^2(x) - 4 = 0$:

Let $\log_{2}(x) = y$, so $\log_{4}(x) = \frac{y}{2}$. Substitute: $\left(\frac{y}{2}\right)^2 + 3y^2 - 4 = 0.$

Simplify and solve for $y$, then back-substitute to find $x$.

Would you like detailed steps for a specific equation or inequality? Let me know!
Here are 5 related questions to expand your understanding:

1. How do you convert between logarithmic and exponential forms?
2. What happens when the base of a logarithm is a fraction?
3. How do you solve compound logarithmic equations?
4. What are the properties of logarithmic functions in inequalities?
5. How can you use substitution to simplify logarithmic equations?

Tip: Always check the domain of logarithmic equations to ensure arguments remain positive.