We can solve the expression $8\log27 \times 3\log4 + 2\log16$ by using properties of logarithms and simplifying step by step.

Step 1: Simplify the logarithms using the exponent rule

The logarithmic property $a\log b = \log b^a$ allows us to move the coefficients inside the logarithm as exponents. Let's apply this property:

$8\log 27 = \log 27^8$ $3\log 4 = \log 4^3$ $2\log 16 = \log 16^2$

Now we have:

$\log 27^8 \times \log 4^3 + \log 16^2$

Step 2: Simplify the expressions

Now, compute the powers:

$27^8 = (3^3)^8 = 3^{24}$ $4^3 = (2^2)^3 = 2^6$ $16^2 = (2^4)^2 = 2^8$

Substituting these into the expression:

$\log 3^{24} \times \log 2^6 + \log 2^8$

Step 3: Use the logarithmic power rule

Using the logarithmic rule $\log a^b = b\log a$, we simplify further:

$24\log 3 \times 6\log 2 + 8\log 2$

Step 4: Multiply the terms

Now, calculate the multiplication:

$(24\log 3) \times (6\log 2) = 144 (\log 3 \times \log 2)$ So, the expression becomes:

$144 (\log 3 \times \log 2) + 8\log 2$

This is the simplified form.

If you want further steps or specific numerical values for the logs, let me know.

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Would you like more details or have any questions?
Here are some related questions to explore:

1. How do you evaluate logarithms with different bases?
2. What are some common properties of logarithms?
3. Can logarithms be negative?
4. How do logarithms relate to exponential functions?
5. How can you solve equations involving logarithms?

Tip: Always check for opportunities to apply logarithmic properties early in simplifications, such as the power, product, or quotient rules.