Question

What is the solution to the expression (4log_3)(4log_6)/(4log_9)(8log_2) + (4log_9)(8log_3)?

a) log_3(2)

b) log_3(4)

c) log_3(6)

d) log_3(8)

Answer 1

The expression using the logarithm properties simplifies to 32, which is not in logarithmic form. It seems there is a mistake in the provided options or the original expression.

Step-by-step explanation:

The solution to the expression (4log_3)(4log_6)/(4log_9)(8log_2) + (4log_9)(8log_3) can be simplified using properties of logarithms. First, let's recall a few properties:

• Logarithm of a quotient: The logarithm of a number resulting from the division of two numbers is the difference between the logarithms of the two numbers (log(a/b) = log(a) - log(b)).
• Logarithm of a power: The logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number (log(a^n) = n*log(a)).

Applying these properties to simplify step by step:

1. Simplify each term using the property of logarithm of a power: (4log_3)(4log_6)/(4log_9)(8log_2) = 16*log_3(6) - 32*log_9(2) = 16*log_3(2*3) - 32*log_9(2)
2. Expand the logarithm using the property of logarithm of a product: 16*(log_3(2) + log_3(3)) - 32*log_9(2)
3. Recognize that log_3(3) is 1 and convert log_9 to base 3: 16*(log_3(2) + 1) - 32*(1/2*log_3(2))
4. Simplify the expression: 16*log_3(2) + 16 - 16*log_3(2)
5. Now address the second part of the expression: (4log_9)(8log_3) = 32*log_9(3) = 32*(1/2)
6. Add the results of the two parts of the expression together: 16 + 16 - 16*log_3(2) + 16
7. The terms involving log_3(2) cancel out, leaving 32 as the final answer, which is not log-based and does not match any of the provided options.

Given the answer is not in logarithmic form, it appears there may be a mistake in the provided options or the original expression. The student should review the expression and the answer choices.