Let’s condense the given logarithmic expressions step by step:

1. $\log_3 5 + \log_3 2$

Using the logarithmic property $\log_b A + \log_b B = \log_b (A \cdot B)$: $\log_3 5 + \log_3 2 = \log_3 (5 \cdot 2) = \log_3 10$

2. $\log_2 R + \log_2 S - 2 \log_2 T$

Using the properties:

• $\log_b A + \log_b B = \log_b (A \cdot B)$
• $n \log_b A = \log_b (A^n)$: $\log_2 R + \log_2 S - 2 \log_2 T = \log_2 (R \cdot S) - \log_2 (T^2)$ Using $\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)$: $\log_2 R + \log_2 S - 2 \log_2 T = \log_2 \left(\frac{R \cdot S}{T^2}\right)$

3. $\log_5 (x^2 - 1) - \log_5 (x - 1)$

Using $\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)$: $\log_5 (x^2 - 1) - \log_5 (x - 1) = \log_5 \left(\frac{x^2 - 1}{x - 1}\right)$ Since $x^2 - 1 = (x - 1)(x + 1)$, the expression simplifies to: $\log_5 \left(\frac{(x - 1)(x + 1)}{x - 1}\right) = \log_5 (x + 1)$

4. $4 \ln (x - 3) + \frac{1}{2} \ln x$

Using $n \ln A = \ln (A^n)$: $4 \ln (x - 3) + \frac{1}{2} \ln x = \ln ((x - 3)^4) + \ln (x^{1/2})$ Using $\ln A + \ln B = \ln (A \cdot B)$: $\ln ((x - 3)^4) + \ln (x^{1/2}) = \ln \left((x - 3)^4 \cdot x^{1/2}\right)$ This simplifies to: $\ln \left(x^{1/2} (x - 3)^4\right)$

5. $\ln x - \ln y + \ln z$

Using $\ln A - \ln B = \ln \left(\frac{A}{B}\right)$ and $\ln A + \ln B = \ln (A \cdot B)$: $\ln x - \ln y + \ln z = \ln \left(\frac{x}{y}\right) + \ln z$ $\ln \left(\frac{x}{y}\right) + \ln z = \ln \left(\frac{x}{y} \cdot z\right)$ This simplifies to: $\ln \left(\frac{x \cdot z}{y}\right)$

Final Results:

1. $\log_3 10$
2. $\log_2 \left(\frac{R \cdot S}{T^2}\right)$
3. $\log_5 (x + 1)$
4. $\ln \left(x^{1/2} (x - 3)^4\right)$
5. $\ln \left(\frac{x \cdot z}{y}\right)$

Would you like additional steps or explanations? Here are five questions to extend your understanding:

1. What are the key logarithmic rules used in condensing these expressions?
2. How does $\ln$ relate to $\log$, and when are they interchangeable?
3. Why do logarithmic properties require the same base for addition/subtraction?
4. What is the significance of condensing logarithmic expressions in problem-solving?
5. How can these properties be used to simplify complex equations?

Tip: Always check if terms can be factored or simplified before applying logarithmic properties!