The problem you uploaded involves finding the exact value of two expressions:

(a) $4 \left( \log_4 7 + \log_4 4 \right)$

Using logarithmic rules, we can simplify the expression step by step:

1. Use the property $\log_b b = 1$. So, $\log_4 4 = 1$.

   This simplifies the expression to: $4 \left( \log_4 7 + 1 \right)$

2. Now distribute the 4: $4 \cdot \log_4 7 + 4 \cdot 1 = 4 \log_4 7 + 4$

3. Using the logarithmic property $a \log_b x = \log_b (x^a)$, the term $4 \log_4 7$ can be rewritten as: $\log_4 (7^4) = \log_4 2401$

Thus, the expression becomes: $\log_4 2401 + 4$ This is the simplified form.

(b) $e^{6 \ln(2)}$

To simplify this, we can use the logarithmic identity $e^{\ln(x)} = x$, which gives: $e^{6 \ln(2)} = 2^6 = 64$

So, the exact value of the second expression is: $64$

Would you like more details on any step or further clarification?

Follow-up Questions:

1. What are the logarithmic properties you can use to simplify expressions?
2. How would the answer change if the base of the logarithm in part (a) were different?
3. What is the general relationship between logarithms and exponentials?
4. How do logarithmic identities help in solving complex logarithmic expressions?
5. Can you convert part (a) into a decimal approximation using logarithmic calculators?

Tip: Always apply logarithmic identities first when simplifying expressions involving logs to avoid unnecessary complexity.