To solve for $x$ in the equation

$\log_3 x + \log_3 10 = \log_3 5 + \log_3 6 - \log_3 2,$

we can use the properties of logarithms to simplify and solve the equation.

Step 1: Use the property $\log_a b + \log_a c = \log_a (bc)$

First, apply the property to combine the logarithms on each side:

$\log_3 (x \cdot 10) = \log_3 (5 \cdot 6) - \log_3 2$

This simplifies to:

$\log_3 (10x) = \log_3 (30) - \log_3 2$

Step 2: Use the property $\log_a b - \log_a c = \log_a \left(\frac{b}{c}\right)$

Now, apply the property to the right side of the equation:

$\log_3 (10x) = \log_3 \left(\frac{30}{2}\right)$

Simplifying the fraction:

$\log_3 (10x) = \log_3 (15)$

Step 3: Since $\log_3 a = \log_3 b \Rightarrow a = b$

We now have:

$10x = 15$

Step 4: Solve for $x$

Divide both sides by 10:

$x = \frac{15}{10} = 1.5$

So, the value of $x$ is $\boxed{1.5}$.

Would you like more details or have any questions? Here are some related problems you might find useful:

1. Solve $\log_2 (x + 3) + \log_2 (x - 1) = 3$.
2. Find $x$ if $\log_4 x + \log_4 (x - 1) = 1$.
3. Solve $\log_5 (2x + 3) = 2 - \log_5 (x - 1)$.
4. Determine $x$ for $\log_3 (x^2 - 4) = 2$.
5. If $\log_2 (3x + 1) = \log_2 (x + 5)$, find $x$.

Tip: Remember that the properties of logarithms can greatly simplify complex logarithmic equations, making them easier to solve.