What is the true solution to 3 ln 2+ln 8=2ln(4x)

x = 2

Step-by-step explanation:

Using the rules of logarithms

• log ⇔ nlog x

• log x + log y ⇔ log(xy)

• log x = log y ⇒ x = y

Given

3 ln2 + ln8 = 2ln(4x)

ln2³ + ln8 = ln(4x)²

ln8 + ln8 = ln16x²

ln(8 × 8) = ln16x²

ln64 = ln16x², hence

16x² = 64 ( divide both sides by 16 )

x² = 4 ( take the square root of both sides )

x = = 2

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.