Let's simplify each expression step by step.

Part a:

The given expression is: $2 \log_2 \sqrt{5} + \frac{1}{3} \log_2 550 - \log_2 10 + \log_2 360.$

1. Simplify $2 \log_2 \sqrt{5}$: $2 \log_2 \sqrt{5} = 2 \cdot \frac{1}{2} \log_2 5 = \log_2 5.$

2. Simplify $\frac{1}{3} \log_2 550$: $\frac{1}{3} \log_2 550 = \log_2 550^{1/3}.$

3. Combine the logs using the properties of logarithms: The expression becomes: $\log_2 5 + \log_2 550^{1/3} - \log_2 10 + \log_2 360.$ Using the logarithm addition and subtraction properties: $\log_2 \left( \frac{5 \cdot 550^{1/3} \cdot 360}{10} \right).$

4. Simplify the argument: $\frac{5 \cdot 550^{1/3} \cdot 360}{10} = 180 \cdot 550^{1/3}.$ So, the simplified form is: $\log_2 (180 \cdot 550^{1/3}).$

Part b:

The given expression is: $\log_{100} 100 + \log_{500} 500 - \log_{1000} 1000.$

1. Simplify each logarithm:

   • $\log_{100} 100 = 1$, since $100^1 = 100$.
   • $\log_{500} 500 = 1$, since $500^1 = 500$.
   • $\log_{1000} 1000 = 1$, since $1000^1 = 1000$.

2. Combine the results: $1 + 1 - 1 = 1.$

Summary:

• Part a: $\log_2 (180 \cdot 550^{1/3})$.
• Part b: 1.

Would you like more details or have any questions about these solutions?

Related Questions:

1. How do logarithmic properties simplify complex logarithmic expressions?
2. What are some other common logarithmic simplification techniques?
3. How does the change of base formula apply to these problems?
4. Can logarithms with different bases be directly combined or simplified?
5. What is the significance of understanding logarithmic functions in real-world applications?

Tip:

Always remember the properties of logarithms: $\log_b (xy) = \log_b x + \log_b y$ and $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$, as these are essential in simplifying expressions.