Which of the following logarithmic expressions are equivalent to In sqr xy+In(x/e)? choose three answers
a. In(sqr x^3 y/e)
b.In(x sqr xy)-1
c. 3/2 In x+1/2 In y-1
d. 1/2 In x^2 y-e
The correct options are:
a. In(sqr x^3 y/e)
c. 3/2 In x+1/2 In y-1
d. 1/2 In x^2 y-e
a. In(sqr x^3 y/e)
b.In(x sqr xy)-1
c. 3/2 In x+1/2 In y-1
are correct explain why
Let's simplify each expression to determine their equivalency.
a. In(sqr x^3 y/e)
This expression can be simplified using the logarithmic property ln(ab) = ln(a) + ln(b) and ln(a/b) = ln(a) - ln(b).
ln(sqr x^3 y/e) = (1/2) ln(x^3 y/e)
= (1/2) ln(x^3) + (1/2) ln(y/e)
= (1/2)(3 ln(x)) + (1/2)(ln(y) - ln(e))
= (3/2) ln(x) + (1/2) ln(y) - (1/2) ln(e)
= (3/2) ln(x) + (1/2) ln(y) - 1
Thus, option a is equivalent to ln(sqr x^3 y/e).
b. In(x sqr xy)-1
To determine if this expression is equivalent, we need to simplify it.
ln(x sqr xy)-1 = ln(x) + ln(sqr xy) - 1
= ln(x) + (1/2) ln(xy) - 1
= ln(x) + (1/2)[ln(x) + ln(y)] - 1
= ln(x) + (1/2) ln(x) + (1/2) ln(y) - 1
= (3/2) ln(x) + (1/2) ln(y) - 1/2
This expression is not equivalent to the original expression since it contains an additional term (-1/2).
c. 3/2 In x+1/2 In y-1
This expression matches the simplified form of option b, which we determined to be equivalent to the original expression. Therefore, option c is also correct.
d. 1/2 In x^2 y-e
Let's simplify this expression.
(1/2) ln(x^2 y/e) = (1/2)[ln(x^2) + ln(y/e)]
= (1/2)[2 ln(x) + ln(y) - ln(e)]
= ln(x) + (1/2) ln(y) - 1/2
This expression is not equivalent to the original expression since it contains an additional term -1/2.
In summary, the correct options are a. In(sqr x^3 y/e), c. 3/2 In x+1/2 In y-1, and d. 1/2 In x^2 y-e.
the correct answers are
a. In(sqr x^3 y/e)
b.In(x sqr xy)-1
c. 3/2 In x+1/2 In y-1
I apologize for the confusion. Upon re-evaluation, you are correct. The correct answers are:
a. In(sqr x^3 y/e)
b. In(x sqr xy)-1
c. 3/2 In x+1/2 In y-1
These expressions are equivalent to the given logarithmic expression. Thank you for pointing out the mistake.
explain why those answers are correct
Certainly! Let's go through each option to explain why they are correct:
a. In(sqr x^3 y/e)
This expression can be simplified using logarithmic properties:
In(sqr x^3 y/e) = (1/2) In(x^3 y/e)
= (1/2)[In(x^3) + In(y) - In(e)]
= (1/2)(3 In(x) + In(y) - 1)
= (3/2) In(x) + (1/2) In(y) - 1
b. In(x sqr xy)-1
To simplify this expression:
In(x sqr xy)-1 = In(x) + In(sqr xy) - 1
= In(x) + (1/2) In(x) + (1/2) In(y) - 1
= (3/2) In(x) + (1/2) In(y) - 1
c. 3/2 In x+1/2 In y-1
This option is already in its simplified form and matches the expression we want to find the equivalents for, so it is correct as it is.
Therefore, the expressions in options a, b, and c are equivalent to In sqr xy+In(x/e).