To simplify the given expression, $\log_4 5^5 + \log_4 25$, we use logarithmic properties:

Step 1: Apply the Product Rule of Logarithms

The sum of two logarithms with the same base can be expressed as the logarithm of the product of their arguments: $\log_a b + \log_a c = \log_a (b \cdot c)$

Here: $\log_4 5^5 + \log_4 25 = \log_4 (5^5 \cdot 25)$

Step 2: Simplify the Argument

Since $25 = 5^2$, substitute $25$ into the product: $\log_4 (5^5 \cdot 5^2) = \log_4 (5^{5+2})$

Step 3: Combine Exponents

$\log_4 (5^{5+2}) = \log_4 (5^7)$

Thus, the simplified form is: $\log_4 5^7$

Step 4: Match with the Options

The correct answer is: D. $\log_4 5^7$

Do you want further explanation or have any other questions? Here are some related questions for practice:

1. What is the logarithm property for division of two logarithmic terms?
2. How would you rewrite $\log_a b^n$ using the power rule?
3. What is the difference between common logarithms and natural logarithms?
4. How can logarithmic equations be used to solve exponential equations?
5. If $\log_a b = x$, what does $a^x = b$ represent?

Tip: Always check if logarithmic expressions can be simplified using the product, quotient, or power rules before solving.