https://file.aiquickdraw.com/mathbot/file/1725379216518k9ip766b.jpg

To show that the expression \(\log 4 \log 40 + \log 5 \log 16\) simplifies to \(\log 64\), let's proceed step by step using properties of logarithms.

### Step 1: Simplify the Expression

The given expression is:

\[
\log 4 \log 40 + \log 5 \log 16
\]

First, let's rewrite the numbers in terms of their prime factors:

- \(4 = 2^2\)
- \(16 = 2^4\)
- \(40 = 2^3 \times 5\)

Now, use the logarithm power rule \(\log(a^b) = b\log(a)\):

- \(\log 4 = \log (2^2) = 2 \log 2\)
- \(\log 16 = \log (2^4) = 4 \log 2\)
- \(\log 40 = \log (2^3 \times 5) = \log (2^3) + \log 5 = 3 \log 2 + \log 5\)

### Step 2: Substitute and Expand

Substitute these values into the original expression:

\[
\log 4 \log 40 = (2 \log 2)(3 \log 2 + \log 5) = 6 (\log 2)^2 + 2 \log 2 \log 5
\]

\[
\log 5 \log 16 = \log 5 (4 \log 2) = 4 \log 5 \log 2
\]

### Step 3: Combine Like Terms

Now combine the terms:

\[
\log 4 \log 40 + \log 5 \log 16 = 6 (\log 2)^2 + 2 \log 2 \log 5 + 4 \log 5 \log 2
\]

Notice that \(2 \log 2 \log 5 + 4 \log 5 \log 2 = 6 \log 2 \log 5\). So, the expression becomes:

\[
6 (\log 2)^2 + 6 \log 2 \log 5
\]

### Step 4: Factor and Simplify

Factor out the common factor of 6:

\[
6 (\log 2)^2 + 6 \log 2 \log 5 = 6(\log 2 (\log 2 + \log 5))
\]

Using the logarithm addition rule, \(\log a + \log b = \log(ab)\):

\[
6(\log 2 \log(2 \times 5)) = 6(\log 2 \log 10)
\]

Now, using the change of base formula, we recognize \(\log 10 = 1\) (since \(\log_{10} 10 = 1\)):

\[
6(\log 2 \cdot 1) = 6 \log 2
\]

Since \(64 = 2^6\), we have:

\[
\log 64 = \log (2^6) = 6 \log 2
\]

Thus, the expression \(\log 4 \log 40 + \log 5 \log 16\) indeed simplifies to \(\log 64\).

### Conclusion

Therefore, we have shown that \(\log 4 \log 40 + \log 5 \log 16 = \log 64\).

---

Would you like further details or have any questions?

Here are some related questions you might find interesting:

1. How can we use logarithm properties to simplify complex expressions?
2. What are some common logarithmic identities useful for solving problems?
3. How can we prove \(\log_{a}(b \times c) = \log_{a}(b) + \log_{a}(c)\)?
4. Why is the logarithm of 10 equal to 1 in base 10?
5. What happens if we change the base of logarithms in an expression?

**Tip:** Remember that logarithm properties such as the product, quotient, and power rules are powerful tools to simplify and solve logarithmic equations.

Learn how to simplify the logarithmic expression log 4 log 40 + log 5 log 16 to log 64 using key logarithmic rules and properties. This step-by-step guide is perfect for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation