To solve for $\log_{12} 36$ using the given approximations $\log_{12} 3 \approx 0.4421$ and $\log_{12} 7 \approx 0.7831$, we can express 36 as a product of the numbers whose logarithms we know:

$36 = 3^2 \times 4 = 3^2 \times (7 \times \frac{3}{7}) = 3^2 \times 7.$

Using the logarithmic properties:

$\log_{12} 36 = \log_{12} (3^2 \times 7).$

Now apply the product and power rule of logarithms:

$\log_{12} (3^2 \times 7) = \log_{12} 3^2 + \log_{12} 7.$

This simplifies to:

$\log_{12} 36 = 2 \log_{12} 3 + \log_{12} 7.$

Substituting the approximated values:

$\log_{12} 36 \approx 2(0.4421) + 0.7831 = 0.8842 + 0.7831 = 1.6673.$

Thus, $\log_{12} 36 \approx 1.6673$.

Would you like any additional details or clarifications?

Here are 5 related questions to expand on this concept:

1. How does the change of base formula work for logarithms?
2. Can you apply logarithmic rules to divide numbers?
3. What are the main properties of logarithms that simplify expressions?
4. How do you approximate logarithms for numbers not directly given?
5. How does the relationship between exponents and logarithms assist in solving equations?

Tip: Always remember to use the logarithmic properties—product, quotient, and power rules—when breaking down complex expressions.