To solve the expression $\log_2(6) - \log_2(12) + \log_2(32)$, we can use the properties of logarithms, specifically the subtraction and addition rules:

1. Subtraction Property: $\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)$
2. Addition Property: $\log_b(x) + \log_b(y) = \log_b(x \cdot y)$

Let's apply these properties step by step.

Step 1: Combine $\log_2(6) - \log_2(12)$

Using the subtraction property: $\log_2(6) - \log_2(12) = \log_2\left(\frac{6}{12}\right) = \log_2\left(\frac{1}{2}\right)$

Step 2: Simplify $\log_2(32)$

Since $32 = 2^5$, we can rewrite $\log_2(32)$ as: $\log_2(32) = 5$

Step 3: Combine $\log_2\left(\frac{1}{2}\right) + \log_2(32)$

Now we add the results from Step 1 and Step 2. Using the addition property: $\log_2\left(\frac{1}{2}\right) + \log_2(32) = \log_2\left(\frac{1}{2} \cdot 32\right)$

Step 4: Simplify $\frac{1}{2} \cdot 32$

$\frac{1}{2} \cdot 32 = 16$

So, we have: $\log_2(16)$

Step 5: Simplify $\log_2(16)$

Since $16 = 2^4$, we can rewrite $\log_2(16)$ as: $\log_2(16) = 4$

Final Answer

$\log_2(6) - \log_2(12) + \log_2(32) = 4$

Would you like a breakdown of these properties or have questions on any steps?

Here are 5 related questions for further exploration:

1. What would happen if the base were changed from 2 to 10?
2. How would you approach $\log_2(8) - \log_2(4) + \log_2(64)$?
3. Why is $\log_b(x^y) = y \cdot \log_b(x)$ a useful property in simplifying expressions?
4. Can you find the result of $\log_2(48) - \log_2(6) + \log_2(4)$?
5. How do you compute $\log_2(1/4)$ using the properties of logarithms?

Tip: Always check if terms in a logarithmic expression can be simplified with powers or fractions before calculating.