To solve the logarithmic equation $\log 5x = \log 2 + \log (x+5)$, we can use the properties of logarithms.

First, apply the property $\log a + \log b = \log (ab)$:

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

Since the logs are equal, the arguments must also be equal:

$5x = 2(x+5)$

Now, solve for $x$:

$5x = 2x + 10$

Subtract $2x$ from both sides:

$3x = 10$

Divide both sides by 3:

$x = \frac{10}{3}$

So, the solution is $x = \frac{10}{3}$.