We are given the expression:

$\log_3{27} + \log_3{6} - \log_3{2}$

Let's simplify this step by step.

Step 1: Simplify $\log_3{27}$

Since $27 = 3^3$, we can simplify:

$\log_3{27} = 3$

Step 2: Apply the logarithmic property for addition

We can combine the logarithms $\log_3{6}$ and $-\log_3{2}$ using the logarithmic property:

$\log_b{x} - \log_b{y} = \log_b{\left(\frac{x}{y}\right)}$

So,

$\log_3{6} - \log_3{2} = \log_3{\left(\frac{6}{2}\right)} = \log_3{3}$

Step 3: Simplify $\log_3{3}$

Since $\log_3{3} = 1$, we can now simplify the expression:

$\log_3{6} - \log_3{2} = 1$

Step 4: Final calculation

Now we add the two simplified terms:

$\log_3{27} + (\log_3{6} - \log_3{2}) = 3 + 1 = 4$

Thus, the simplified value of the expression is:

$\boxed{4}$

Would you like more details or have any questions?

Here are 5 related questions you might find helpful:

1. How do logarithmic addition and subtraction rules work in general?
2. What is the change of base formula for logarithms?
3. How do you solve equations involving logarithms and exponents?
4. Can you express $\log_b{a^x}$ in simpler terms?
5. How do logarithms relate to exponential growth and decay?

Tip: When simplifying logarithmic