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The image asks to express the following logarithmic equations **without logarithms**. Let's work through each equation step by step.

### a) \( \log x = \log P + 2 \log Q - \log k - 3 \)

Using logarithmic properties, we can simplify and eliminate the logarithms:

- \( \log P + 2 \log Q = \log (P Q^2) \)
- \( \log P + 2 \log Q - \log k = \log \left( \frac{P Q^2}{k} \right) \)

So, the equation becomes:

\[
\log x = \log \left( \frac{P Q^2}{k} \right) - 3
\]

This means:

\[
\log x = \log \left( \frac{P Q^2}{k} \right) + \log \left( \frac{1}{10^3} \right)
\]

Thus:

\[
\log x = \log \left( \frac{P Q^2}{k} \cdot \frac{1}{1000} \right)
\]

Finally, removing the logarithms:

\[
x = \frac{P Q^2}{k \cdot 1000}
\]

### b) \( \log R = 1 + \frac{1}{3} \log M + 3 \log S \)

Again, apply logarithmic properties:

- \( \frac{1}{3} \log M = \log \left( M^{1/3} \right) \)
- \( 3 \log S = \log (S^3) \)

Thus, the equation becomes:

\[
\log R = 1 + \log \left( M^{1/3} \right) + \log (S^3)
\]

Using \( 1 = \log 10 \):

\[
\log R = \log 10 + \log \left( M^{1/3} \right) + \log (S^3)
\]

This simplifies to:

\[
\log R = \log \left( 10 \cdot M^{1/3} \cdot S^3 \right)
\]

Removing the logarithms:

\[
R = 10 \cdot M^{1/3} \cdot S^3
\]

### c) \( \ln p = \frac{1}{2} \ln (Q + 1) - 3 \ln R + z \)

First, express the terms with logarithmic properties:

- \( \frac{1}{2} \ln (Q + 1) = \ln \left( (Q + 1)^{1/2} \right) \)
- \( -3 \ln R = \ln \left( \frac{1}{R^3} \right) \)

So, the equation becomes:

\[
\ln p = \ln \left( (Q + 1)^{1/2} \right) + \ln \left( \frac{1}{R^3} \right) + z
\]

Now, combine the logarithmic terms:

\[
\ln p = \ln \left( \frac{(Q + 1)^{1/2}}{R^3} \right) + z
\]

Removing the natural logarithm:

\[
p = \frac{(Q + 1)^{1/2}}{R^3} \cdot e^z
\]

### Summary of Final Results:
1. \( x = \frac{P Q^2}{k \cdot 1000} \)
2. \( R = 10 \cdot M^{1/3} \cdot S^3 \)
3. \( p = \frac{(Q + 1)^{1/2}}{R^3} \cdot e^z \)

Would you like more details or explanations on any step?

Here are 5 related questions for further practice:
1. How would you express \( \log a + 2 \log b - \log c \) without logarithms?
2. How do you simplify \( \log (x^2y) \)?
3. What is the value of \( \log x = 1 + \log y \)?
4. Can you explain the property \( \log (a \cdot b) = \log a + \log b \)?
5. How do you solve \( \ln x = 2 \ln y - \ln z \)?

**Tip:** Remember that \( \log a^b = b \log a \), a key logarithmic rule for simplifying terms.

Express the following logarithmic equations without logarithms.

Learn how to express logarithmic equations without logarithms using logarithmic properties and exponentiation. This exercise covers key algebraic concepts suitable for high school students.

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