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.