The image contains a set of logarithmic problems. Let's solve the problems shown:

Question a:

$\log_3 243$ We need to find $\log_3 243$, which means we are solving for $x$ in the equation: $3^x = 243$ We know that $243 = 3^5$, so: $\log_3 243 = 5$

Question b:

$\log_5 25\sqrt{5}$ This can be simplified as: $\log_5 \left(25 \times \sqrt{5}\right) = \log_5 \left(5^2 \times 5^{1/2}\right)$ Using the logarithmic property $\log_b (a \times c) = \log_b a + \log_b c$, we get: $\log_5 \left(5^2 \times 5^{1/2}\right) = \log_5 5^2 + \log_5 5^{1/2}$ Since $\log_5 5^n = n$, we can simplify further: $\log_5 5^2 + \log_5 5^{1/2} = 2 + \frac{1}{2} = 2.5$

Final Answers:

a. $\log_3 243 = 5$
b. $\log_5 25\sqrt{5} = 2.5$

Would you like further explanations on logarithmic properties or any other details?

Here are 5 related questions:

1. How do you express logarithmic equations in exponential form?
2. What are the properties of logarithms, and how are they applied in solving problems?
3. Can you solve $\log_2 32$ using logarithmic properties?
4. How would you solve a logarithmic equation where the base is unknown?
5. What is the change of base formula for logarithms?

Tip: Understanding the fundamental properties of logarithms, such as the product, quotient, and power rules, can greatly simplify solving logarithmic expressions.