To write $\log \sqrt{3x}$ in expanded form, we can use the properties of logarithms.

Given the expression:

$\log \sqrt{3x}$

First, recall that $\sqrt{3x}$ can be written as $(3x)^{\frac{1}{2}}$.

So, we have:

$\log \sqrt{3x} = \log (3x)^{\frac{1}{2}}$

Next, use the power rule of logarithms, which states $\log a^b = b \log a$:

$\log (3x)^{\frac{1}{2}} = \frac{1}{2} \log (3x)$

Then, use the product rule of logarithms, which states $\log (ab) = \log a + \log b$:

$\frac{1}{2} \log (3x) = \frac{1}{2} (\log 3 + \log x)$

Distribute the $\frac{1}{2}$ to both terms inside the parentheses:

$\frac{1}{2} (\log 3 + \log x) = \frac{1}{2} \log 3 + \frac{1}{2} \log x$

Thus, the expanded form of $\log \sqrt{3x}$ is:

$\log \sqrt{3x} = \frac{1}{2} \log 3 + \frac{1}{2} \log x$

Would you like more details or have any questions?

Here are some more questions you might have:

1. How do you use other logarithm properties to expand expressions?
2. What are some examples of using the power rule for logarithms?
3. Can you explain the change of base formula for logarithms?
4. How do you simplify logarithmic expressions with different bases?
5. What is the relationship between logarithms and exponents?

Tip: When expanding logarithmic expressions, remember to apply the logarithm properties step-by-step to simplify correctly.