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The given expression is:

\[
\frac{{^4\log 25 \times ^5\log 64 + ^5\log 25}}{{^5\log 75 - ^5\log 3}}
\]

Let's simplify this step-by-step:

### Step 1: Simplify the logs with different bases

Using the change of base formula:

\[
^a\log b = \frac{\log b}{\log a}
\]

So, let's simplify each logarithmic term:

1. \( ^4\log 25 = \frac{\log 25}{\log 4} \)
2. \( ^5\log 64 = \frac{\log 64}{\log 5} \)
3. \( ^5\log 25 = \frac{\log 25}{\log 5} \)
4. \( ^5\log 75 = \frac{\log 75}{\log 5} \)
5. \( ^5\log 3 = \frac{\log 3}{\log 5} \)

### Step 2: Substitute these values into the expression

Substituting the simplified terms into the expression:

\[
\frac{\left(\frac{\log 25}{\log 4} \times \frac{\log 64}{\log 5}\right) + \frac{\log 25}{\log 5}}{\frac{\log 75}{\log 5} - \frac{\log 3}{\log 5}}
\]

### Step 3: Combine the fractions in the numerator and simplify

1. Combine the terms in the numerator:
   \[
   \frac{\log 25 \times \log 64}{\log 4 \times \log 5} + \frac{\log 25}{\log 5}
   \]

2. Simplify the denominator:
   \[
   \frac{\log 75 - \log 3}{\log 5} = \frac{\log \left(\frac{75}{3}\right)}{\log 5} = \frac{\log 25}{\log 5}
   \]

### Step 4: Substitute and further simplify

Now substitute the simplified denominator into the expression:

\[
\frac{\frac{\log 25 \times \log 64}{\log 4 \times \log 5} + \frac{\log 25}{\log 5}}{\frac{\log 25}{\log 5}}
\]

Simplify this by factoring \(\frac{\log 25}{\log 5}\):

\[
= \frac{\log 25}{\log 5} \left(\frac{\log 64}{\log 4 \times \log 5} + 1\right) \div \frac{\log 25}{\log 5}
\]

This simplifies to:

\[
= \frac{\log 64}{\log 4 \times \log 5} + 1
\]

Finally, knowing that \(\log 64 = 6 \log 4\):

\[
= \frac{6 \log 4}{\log 4 \times \log 5} + 1 = 6 + 1 = 7
\]

Thus, the simplified value of the expression is:

\[
\boxed{7}
\]

Would you like more details, or have any questions about the steps?

### Related Questions:
1. How can you simplify logarithmic expressions with different bases more effectively?
2. What are the properties of logarithms, and how do they apply here?
3. How does the change of base formula work, and when is it useful?
4. What happens if the bases of logarithms are irrational or complex?
5. How does simplifying log expressions help in solving equations?

### Tip:
Always look for opportunities to factor and simplify common terms in complex expressions to make calculations easier!

4log25 × 5log64 + 5log25 / 5log75 - 5log3

Learn how to simplify complex logarithmic expressions using the change of base formula and logarithmic properties. This example covers the step-by-step process of solving 4log25 × 5log64 + 5log25 / 5log75 - 5log3.

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