Expand the logarithm (10 Points!)

ln 2x^3 / (x + 9)^10

I'm assuming you mean ln ( 2x^3 / (x + 9)^10 )

If separated by division, you subtract when you separate it:
ln (2x^3) - ln ( (x + 9)^10 )

With multiplication, you add:
ln 2 + ln x^3 - ln ( (x + 9)^10 )

And with exponents, you "bring them in front":
ln 2 + 3 ln x - 10 ln (x + 9)

And I believe that's as far as you can expand it

To expand the squares, we'll use the formula:

(a+b)^2 = a^2 + 2ab + b^2

We'll expand the square: (2x-3)^2

(2x-3)^2 = (2x)^2 - 2*2x*3 + 3^2

(2x-3)^2 = 4x^2 - 12x + 9

We'll expand (x+2)^2:

(x+2)^2 = x^2 + 2*x*2 + 2^2

(x+2)^2 = x^2 + 4x + 4

We'll re-write the equation:

4x^2 - 12x + 9 + x^2 + 4x + 4 = 10+5x^2

We'll subtract both sides (10+5x^2):

4x^2 - 12x + 9 + x^2 + 4x + 4 - 10 - 5x^2 = 0

We'll combine and eliminate like terms:

-8x + 3 = 0

We'll subtract 3 both sides:

-8x = -3

We'll divide by -8:

x = -3/-8

x = 3/8

x = 0.375

(ln(2 x^3))/(x^10+90 x^9+3645 x^8+87480 x^7+1377810 x^6+14880348 x^5+111602610 x^4+573956280 x^3+1937102445 x^2+3874204890 x+3486784401)

ln [ 2x^3 / (x + 9)^10 ]

= ln (2x^3) - ln (x + 9)^10

= ln 2 + 3 ln x - 10 ln (x + 9 )