To expand the expression $\ln(3x)$ using the properties of logarithms, specifically the product rule, we can break it down as follows:

The product rule of logarithms:

$\ln(ab) = \ln(a) + \ln(b)$ In this case, $a = 3$ and $b = x$, so applying the product rule:

$\ln(3x) = \ln(3) + \ln(x)$

Thus, the expanded form of $\ln(3x)$ is: $\ln(3x) = \ln(3) + \ln(x)$

This is the final expansion.

Would you like further details or have any questions?
Here are 5 related questions to explore further:

1. How do you apply the quotient rule of logarithms?
2. What is the power rule of logarithms, and how is it used?
3. Can you simplify logarithmic expressions involving exponents?
4. How do you solve logarithmic equations like $\ln(3x) = 5$?
5. What is the base change formula for logarithms?

Tip: The product rule for logarithms is helpful when breaking down multiplication inside the logarithm into a sum of simpler logs.