expand log_1/2 (3x^2/2) using the properties and rules for logarithms

A. log_1/2(3)+2log(x)-log_1/2(2)
B. log_1/2(3)+2log_1/2(x)+1
C. 2log_1/2(3x)+1
D. 2log_1/2(3x)-log_1/2(2)

1 answer

To expand log base 1/2 of (3x^2/2), we can use the properties and rules for logarithms.

First, we can use the rule that log base b of (a * c) equals log base b of a plus log base b of c. Applying this rule to our expression, we get:

log base 1/2 of (3) + log base 1/2 of (x^2/2)

Next, we can use another rule that log base b of (a^c) equals c * log base b of a. Applying this rule to the second term of our expression, we get:

log base 1/2 of (3) + (2/2) * log base 1/2 of (x)

Simplifying further, we have:

log base 1/2 of (3) + log base 1/2 of (x)

Finally, we can combine these logarithms using the rule that log base b of a plus log base b of c is equal to log base b of (a * c). Applying this rule to our expression, we get:

log base 1/2 of (3 * x)

Therefore, the expanded form of log base 1/2 of (3x^2/2) is:

2log base 1/2 of (3x)

So the correct answer is C. 2log base 1/2 of (3x) + 1.