Let's break down each of the logarithmic expressions and analyze the correct choices for each problem.

1. Write $\log_b (xy)$ in expanded form:

The property of logarithms that applies here is: $\log_b (xy) = \log_b (x) + \log_b (y)$ This is the product rule for logarithms. Therefore, the correct expanded form is:

• (iv) $\log_b (x) + \log_b (y)$

2. Write $\log_b \left(\frac{x}{y}\right)$ in expanded form:

The property of logarithms that applies here is: $\log_b \left( \frac{x}{y} \right) = \log_b (x) - \log_b (y)$ This is the quotient rule for logarithms. Therefore, the correct expanded form is:

• (ii) $\log_b (x) - \log_b (y)$

3. The expression $\log_b (x^p)$ equals:

The property of logarithms that applies here is: $\log_b (x^p) = p \cdot \log_b (x)$ This is the power rule for logarithms. Therefore, the correct expanded form is:

• (i) $p \cdot \log_b (x)$

4. The statement $\log_b (x + y) = \log_b (x) + \log_b (y)$ is:

This statement is False. The property that $\log_b (x + y) = \log_b (x) + \log_b (y)$ does not hold in general logarithmic identities. Instead, $\log_b$ only works with products, quotients, and powers as shown above. So the correct answer is:

• (ii) False

5. Which of the following equals $\log_2 (x)$?

To convert between different logarithmic bases, you can use the change of base formula: $\log_b (x) = \frac{\log_k (x)}{\log_k (b)}$ Using this formula, $\log_2 (x)$ can be written in terms of base 5 as: $\log_2 (x) = \frac{\log_5 (x)}{\log_5 (2)}$ Thus, the correct answer is:

• (iv) $\frac{\log_5 (x)}{\log_5 (2)}$

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Summary of correct answers:

1. (iv) $\log_b (x) + \log_b (y)$
2. (ii) $\log_b (x) - \log_b (y)$
3. (i) $p \log_b (x)$
4. (ii) False
5. (iv) $\frac{\log_5 (x)}{\log_5 (2)}$

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Would you like further explanations or have any additional questions?

5 Related Questions:

1. How does the logarithmic change of base formula work?
2. Can you provide examples of logarithm properties with real numbers?
3. What are the key differences between natural logarithms and base-10 logarithms?
4. How can logarithms be used to solve exponential equations?
5. How do you simplify expressions with multiple logarithmic rules?

Tip:

Remember that logarithmic properties are powerful tools for simplifying expressions, but always ensure you're applying the correct rule for products, quotients, or powers!