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)
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.
To expand log base 1/2 of (3x^2/2) using the properties and rules for logarithms, we can apply the following rules:
1. log base a (mn) = log base a (m) + log base a (n)
2. log base a (m^n) = n * log base a (m)
3. log base a (1) = 0
Using these rules, let's simplify step by step:
log base 1/2 (3x^2/2)
= log base 1/2 (3) + log base 1/2 (x^2/2) (using rule 1)
= log base 1/2 (3) + (2 * log base 1/2 (x)) - log base 1/2 (2) (using rule 2 twice)
= log base 1/2 (3) + 2 * log base 1/2 (x) - log base 1/2 (2) (simplifying the exponent of x)
= log base 1/2 (3) + 2log(base 1/2) (x) - log base 1/2 (2) (rearranging terms)
Therefore, the expansion of log base 1/2 (3x^2/2) using the properties and rules for logarithms is:
A. log base 1/2 (3) + 2log(base 1/2) (x) - log base 1/2 (2)
To expand the logarithm log_1/2 (3x^2/2) using the properties and rules for logarithms, we can use the following rules:
1. log_a (b * c) = log_a (b) + log_a (c)
2. log_a (b^c) = c * log_a (b)
3. log_a (a) = 1
4. log_a (1) = 0
Taking these rules into consideration, let's expand the given logarithm step by step:
log_1/2 (3x^2/2)
Using the rule 1, we can split log_1/2 (3x^2/2) into two separate logarithms:
= log_1/2 (3) + log_1/2 (x^2/2)
Now, applying rule 2 to the second logarithm:
= log_1/2 (3) + (2/2) * log_1/2 (x)
Simplifying further:
= log_1/2 (3) + log_1/2 (x)
Finally, by using rule 3 and rule 4, we can rewrite the logarithms with a base of 1/2 as negative exponents:
= log_1/2 (3) + log_1/2 (x)
= log_1/2 (3) + log_1/2 (x^1)
= log_1/2 (3) + log_1/2 (2^2 * x^-1)
= log_1/2 (3) + (2 * log_1/2 (2)) - (1 * log_1/2 (x))
Combining like terms in the second logarithm:
= log_1/2 (3) + (2 * 1) - log_1/2 (x)
= log_1/2 (3) + 2 - log_1/2 (x)
Therefore, the expanded form of log_1/2 (3x^2/2) is option A. log_1/2 (3) + 2log(x) - log_1/2 (2).