The problem appears to involve finding $x$ in two different equations involving logarithms. The task is split into two variants:

Variant 1:

$\log x^2 = 2 \log 8 - 2$

Solution for Variant 1:

1. Start with the given equation: $\log x^2 = 2 \log 8 - 2$

2. Use the property of logarithms that $\log x^2 = 2 \log x$, so the equation becomes: $2 \log x = 2 \log 8 - 2$

3. Factor out 2 from the right-hand side: $2 \log x = 2(\log 8 - 1)$

4. Divide both sides by 2: $\log x = \log 8 - 1$

5. Since $\log 8 = \log (2^3) = 3 \log 2$, we get: $\log x = 3 \log 2 - 1$

6. Let’s assume a logarithm base 10, and find the value of $x$.

Variant 2:

$\log x^2 = 2 \log 4 - 2$

Solution for Variant 2:

1. Start with the given equation: $\log x^2 = 2 \log 4 - 2$

2. Use the same property: $\log x^2 = 2 \log x$, so: $2 \log x = 2 \log 4 - 2$

3. Simplify by factoring out 2: $2 \log x = 2(\log 4 - 1)$

4. Divide by 2: $\log x = \log 4 - 1$

5. Since $\log 4 = \log (2^2) = 2 \log 2$, we get: $\log x = 2 \log 2 - 1$

6. Solve for $x$.

Let me know if you need the detailed solutions or have any further questions!

Five Questions to Explore:

1. What are the properties of logarithms that are useful in solving equations like these?
2. How would the equations change if the base of the logarithm was not 10?
3. How can we use a calculator to find the numerical value of $x$?
4. What are some common logarithmic equations and their applications?
5. Can we solve these equations using natural logarithms (base $e$) instead?

Tip:

Remember that logarithmic equations often require the use of properties like the power rule ($\log a^b = b \log a$) and the product rule ($\log(ab) = \log a + \log b$). Familiarizing yourself with these can make solving problems quicker and more intuitive!