We start by using the logarithmic property that allows combining terms:

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

The given equation is:

$\log 3x = \log 2 + \log (x + 5).$

Combine the terms on the right-hand side:

$\log 3x = \log [2(x + 5)].$

Since the logarithmic function is one-to-one, we can equate the arguments of the logarithms:

$3x = 2(x + 5).$

Solve for $x$:

1. Distribute $2$ on the right-hand side: $3x = 2x + 10.$

2. Subtract $2x$ from both sides: $x = 10.$

Verify the solution:

Substitute $x = 10$ back into the original equation:

$\log (3 \cdot 10) = \log 2 + \log (10 + 5).$

This simplifies to:

$\log 30 = \log 2 + \log 15.$

Combine terms on the right: $\log 30 = \log (2 \cdot 15) = \log 30.$

The solution holds true. Thus, the solution is:

$\boxed{x = 10.}$

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Would you like further explanation or a breakdown of any step?

5 Related Questions:

1. What are the key properties of logarithms used to solve equations like this?
2. How do you solve equations involving natural logarithms ($\ln$) instead of $\log$?
3. Can logarithmic equations have extraneous solutions? How do you check for them?
4. What is the difference between $\log x$ and $\ln x$?
5. How do you graph the function $\log x$ and its transformations?

Tip:

Always verify solutions to logarithmic equations by substituting them back into the original equation to ensure the arguments of the logarithms are positive.