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The image you uploaded contains a whiteboard with logarithmic equations written on it. I'll go ahead and extract the problems to help solve them.

Here is a breakdown of the logarithmic equations visible:

1. \(\log x^2 = \frac{1}{4} \log y^2\)
2. \(\log \frac{x^3}{y^2} = \frac{1}{2} \log x^4\)
3. \(\log x - \log y^2 = \frac{1}{3} \log x^6 y\)
4. \(\log \frac{x^2 y^3}{z^4} = 2 \log z\)
5. \(\log \left( \frac{x^3}{y^4} \right) = \log x^4\)
6. \(\log x^2 = 2 \log y - \log z\)
7. \(\log (x + y) = 2 \log z\)

I will proceed by solving these equations systematically:

### Equation 1:
\[
\log x^2 = \frac{1}{4} \log y^2
\]

Using logarithmic rules:

\[
2 \log x = \frac{1}{2} \log y
\]

Simplifying:

\[
4 \log x = \log y
\]

This gives:

\[
x^4 = y
\]

---

### Equation 2:
\[
\log \frac{x^3}{y^2} = \frac{1}{2} \log x^4
\]

Expanding both sides using logarithmic properties:

\[
\log x^3 - \log y^2 = 2 \log x
\]

Simplifying:

\[
3 \log x - 2 \log y = 2 \log x
\]

Subtracting \(2 \log x\) from both sides:

\[
\log x = 2 \log y
\]

Which simplifies to:

\[
x = y^2
\]

---

### Equation 3:
\[
\log x - \log y^2 = \frac{1}{3} \log x^6 y
\]

Simplifying both sides:

\[
\log x - 2 \log y = 2 \log x + \log y
\]

Rearranging terms:

\[
\log x - 2 \log x = 2 \log y + \log y
\]

\[
-\log x = 3 \log y
\]

This gives:

\[
x = \frac{1}{y^3}
\]

---

### Equation 4:
\[
\log \frac{x^2 y^3}{z^4} = 2 \log z
\]

Simplifying:

\[
\log x^2 + \log y^3 - \log z^4 = 2 \log z
\]

\[
2 \log x + 3 \log y - 4 \log z = 2 \log z
\]

Simplifying further:

\[
2 \log x + 3 \log y = 6 \log z
\]

Thus:

\[
x^2 y^3 = z^6
\]

---

### Equation 5:
\[
\log \left( \frac{x^3}{y^4} \right) = \log x^4
\]

This simplifies as:

\[
\log x^3 - \log y^4 = 4 \log x
\]

Simplifying:

\[
3 \log x - 4 \log y = 4 \log x
\]

Subtracting \(3 \log x\) from both sides:

\[
-4 \log y = \log x
\]

Thus:

\[
y = \frac{1}{x^4}
\]

---

### Equation 6:
\[
\log x^2 = 2 \log y - \log z
\]

Simplifying:

\[
2 \log x = 2 \log y - \log z
\]

Dividing by 2:

\[
\log x = \log y - \frac{1}{2} \log z
\]

Thus:

\[
x = \frac{y}{\sqrt{z}}
\]

---

### Equation 7:
\[
\log (x + y) = 2 \log z
\]

Simplifying:

\[
\log (x + y) = \log z^2
\]

Thus:

\[
x + y = z^2
\]

---

Would you like more detailed steps on any specific equation, or is there anything else you'd like to clarify?

### Related Questions:
1. Can you explain the logarithmic properties used in solving the equations?
2. How would you approach solving a logarithmic equation with multiple variables?
3. What other transformations can simplify logarithmic equations?
4. Can we always solve logarithmic equations algebraically, or are there exceptions?
5. How do you interpret the results from logarithmic equations in practical applications?

### Tip:
Always remember to simplify logarithmic expressions by applying properties such as product, quotient, and power rules.

This detailed guide walks through the solution of complex logarithmic equations such as log(x^2) = (1/4) log(y^2) and log(x^3/y^2) = (1/2) log(x^4), using logarithmic rules like the product, quotient, and power rules. Ideal for students in Grades 10-12.

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