Question

Expand using the properties of logarithmslog9(3a^2bc^3)^2​

A) 2 * (log9(3) + 2log9(a) + log9(b) + 6log9(c))
B) 4 * log9(3) + 4log9(a) + 4log9(b) + 8log9(c)
C) 2 * log9(3) + 4log9(a) + 2log9(b) + 12log9(c)
D) 4 * (log9(3) + log9(a) + log9(b) + 2log9(c))

Answer 1

The correct expansion of the logarithmic expression log9(3a^2bc^3)^2 using properties of logarithms is option A: 2 * (log9(3) + 2 * log9(a) + log9(b) + 6 * log9(c)). The properties used include taking exponents out in front of the log and the logarithm of a product being the sum of the logarithms.

Step-by-step explanation:

The question requires us to expand the logarithmic expression log9(3a^2bc^3)^2 using the properties of logarithms. According to the property of exponents in logarithms, which states that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number, we can take the exponent 2 out in front of the log.

log9(3a^2bc^3)^2 = 2 * log9(3a^2bc^3)

Next, using the property that the logarithm of a product of numbers is the sum of the logarithms of those numbers, we can expand further:

2 * log9(3a^2bc^3) = 2 * (log9(3) + log9(a^2) + log9(b) + log9(c^3))

Next, we apply the property of exponents within the logarithms:

2 * (log9(3) + log9(a^2) + log9(b) + log9(c^3)) = 2 * (log9(3) + 2 * log9(a) + log9(b) + 3 * log9(c))

Finally, we can distribute the 2 into each term:

2 * (log9(3) + 2 * log9(a) + log9(b) + 3 * log9(c)) = 2 * log9(3) + 4 * log9(a) + 2 * log9(b) + 6 * log9(c)

Therefore, the correct expansion of the logarithmic expression using the properties of logarithms is option A: 2 * (log9(3) + 2 * log9(a) + log9(b) + 6 * log9(c)).