Let's break down the given equation step by step and prove that:

$\log_4 36 + \frac{1}{2} \log_4 256 - 2 \log_4 48 = -\log_4 4$

Step 1: Simplify the logarithmic terms individually

We'll use properties of logarithms such as:

• $\log_b (mn) = \log_b m + \log_b n$
• $\log_b (m/n) = \log_b m - \log_b n$
• $\log_b (m^p) = p \log_b m$

Term 1: $\log_4 36$

Leave as is for now: $\log_4 36$.

Term 2: $\frac{1}{2} \log_4 256$

Use the power rule: $\frac{1}{2} \log_4 256 = \log_4 (256^{1/2}) = \log_4 16$.

Term 3: $2 \log_4 48$

Use the power rule: $2 \log_4 48 = \log_4 (48^2) = \log_4 2304$.

Step 2: Substitute and simplify the equation

The equation now becomes: $\log_4 36 + \log_4 16 - \log_4 2304 = -\log_4 4$

Step 3: Combine the logarithms using addition and subtraction rules

Use the logarithmic rule $\log_b m + \log_b n = \log_b (m \cdot n)$ and $\log_b m - \log_b n = \log_b \left( \frac{m}{n} \right)$.

$\log_4 \left( 36 \times 16 \right) - \log_4 2304 = -\log_4 4$ $\log_4 576 - \log_4 2304 = -\log_4 4$

Now apply the subtraction rule: $\log_4 \left( \frac{576}{2304} \right) = -\log_4 4$ Simplify the fraction $\frac{576}{2304} = \frac{1}{4}$, so we get: $\log_4 \left( \frac{1}{4} \right) = -\log_4 4$

Step 4: Solve the logarithms

Since $\log_4 \left( \frac{1}{4} \right) = \log_4 4^{-1} = -\log_4 4$, we have: $-\log_4 4 = -\log_4 4$

Thus, both sides are equal, proving the equation is correct.

Conclusion:

$\log_4 36 + \frac{1}{2} \log_4 256 - 2 \log_4 48 = -\log_4 4$

Do you want any further clarification or details?

Here are 5 related questions you might consider:

1. How do we handle logarithms with different bases?
2. What is the change of base formula for logarithms?
3. How can logarithmic identities simplify complex expressions?
4. What are the real-world applications of logarithmic functions?
5. How does the relationship between exponents and logarithms work?

Tip: Always keep logarithmic properties handy, such as the product, quotient, and power rules. They help simplify expressions efficiently.