Express as a single natural logarithm.
ln 16 - ln 8
a. ln 2
b. ln 8
c. ln 128

Answers To The Whole Assignment

1. middle graph

2. middle graph

3.  in e^3 = 3, in e^2y=2y

4. in e^ln1=1, in e^ln5x=5x

5. A. In2

6.  A. In4

7. B. ln a + 2 ln b  C. ln a2 + ln b2 - ln a E. ln ab2

8. C. ln 25x + ln 1  D. ln 25x

9. In 3x= in3+ln x= in In 2a/b= In2+ln a-lnb

10. A. 4 ln 2 + 4 ln x B. ln 4 - 2 ln x + 5 ln y

11. A. x=5 B. x=6

12. C. {8} B. {3}

13.  5

14. Since 0 in ln(3x) - 0 is not a logarithm, the  property of logarithms cannot be used here.The difference shown cannot be written as a quotient of logarithms.The step ln(x2) = ln(3x) - (0) reduces to 

ln(x2) = ln(3x).The possible solutions are 0 and 3, with 0 being extraneous.

3. parallelogram

step-by-step explanation:

we turn each of our sides into direction vectors

ab=b-a=(3 - 7, 1 - 0) = (-4, 1)

bc=c-b=(1,6)

cd=d-c=(4,-1)

da=a-d=(-1,-6)

we see how the opposite sides have negated direction vectors; this is characteristic of a parallelogram.   we have congruent sides ab=cd and bc=da with unequal lengths because 1^2+6^2 is different from (-4)^2+1^2.   so it's not a square or a rhombus.  

we check the dot product of direction vectors of adjacent sides.   we have

ab dot bc = -4(1) + 1(6) = 2

that's not zero so the sides are not perpendicular.   we've ruled out rectangle.

the direction vectors indicate two pairs of congruent parallel sides, so we choose parallelogram over trapezoid.