We can simplify this logarithmic expression step by step.

Given expression:

$\log 2\sqrt{2} + \log \sqrt{3} + \log 18$

Step 1: Apply the logarithmic property:

$\log a + \log b + \log c = \log (a \cdot b \cdot c)$

So, we can combine all the logarithms: $\log (2\sqrt{2} \cdot \sqrt{3} \cdot 18)$

Step 2: Simplify the expression inside the logarithm:

First, simplify the terms inside the logarithm.

• $2\sqrt{2} \cdot \sqrt{3} = 2 \cdot \sqrt{6}$
• $2 \cdot \sqrt{6} \cdot 18 = 36\sqrt{6}$

Thus, the expression becomes: $\log (36\sqrt{6})$

Step 3: Express the product of 36 and $\sqrt{6}$:

We can break this down into: $\log (36) + \log (\sqrt{6})$

Step 4: Use logarithmic properties:

$\log (36) = \log (6^2) = 2\log(6)$ And: $\log (\sqrt{6}) = \frac{1}{2} \log (6)$

Step 5: Combine terms:

$2\log (6) + \frac{1}{2} \log (6) = \frac{5}{2} \log (6)$

Thus, the simplified expression is: $\boxed{\frac{5}{2} \log (6)}$

Would you like more details or have any questions?

Here are 5 related questions you might find useful:

1. How can you expand the logarithmic properties to handle complex expressions?
2. What is the change of base formula for logarithms?
3. How do you simplify logarithmic expressions involving square roots?
4. What are the key logarithmic identities used in algebra?
5. How do you solve equations involving multiple logarithms?

Tip: Always look for ways to combine logarithmic terms using their properties—this often simplifies the problem significantly.