Question

Simplify the logarithmic expression.

3log₁₂² + (1)/(3) log₁₂⁸ - log₁₂⁴
A. log₃₂⁸
B. log₁₂⁶
C. log₁₂⁴
D. log₁₂¹⁶

Answer 1

To simplify the logarithmic expression, we use the properties of logarithms to combine the terms. After simplification, the answer is c. log₁₂⁴.

Step-by-step explanation:

To simplify the given logarithmic expression, we can use the properties of logarithms. The properties we will use are:

1. Product rule: log(ab) = log(a) + log(b)
2. Quotient rule: log(a/b) = log(a) - log(b)

Using these properties, we can simplify the expression step-by-step:

3log₁₂² + (1)/(3) log₁₂⁸ - log₁₂⁴

= log₁₂(2³) + log₁₂(8^(1/3)) - log₁₂(4)

= log₁₂(8) + log₁₂(2) - log₁₂(4)

= 3log₁₂(2) + log₁₂(8) - log₁₂(4)

= 3log₁₂(2) + log₁₂(2³) - log₁₂(2²)

= 3log₁₂(2) + 3log₁₂(2) - 2log₁₂(2)

= 6log₁₂(2) - 2log₁₂(2)

= (6 - 2)log₁₂(2)

= 4log₁₂(2)

The simplified expression is 4log₁₂(2). Therefore, the answer is log₁₂⁴ (option C).