ln(x-6)+ln(x+3)=ln22?

Question

ln(x-6)+ln(x+3)=ln22

its asking to include approx. to the nearest thousandth need help thanks

Answer 1

(x - 6)(x + 3) = 22
x² - 3x - 18 = 22
x² - 3x - 40 = 0
(x - 8) (x + 5) = 0
x = 8 , x = - 5
Accept the +ve value of x = 8

Answer 2

The sum of two logs is the log of their product:
ln (x - 6)(x + 3) = ln 22

The expression after the log on the left must be equal to the same on the right:
(x - 6)(x + 3) = 22

Simplify:
x² -6x +3x -18 = 22
Solve for zero:
x² -6x +3x -18 -22 = 0
x² -3x - 40 = 0
Factor:
(x - 8)(x+5) = 0
Use zero-product property:
x - 8 = 0 or x + 5 = 0
x = 8 or x = -5

Substitute to check:
ln (8-6) + ln(8+3) = ln 22
ln 2 + ln 11 = ln 22
0.693147 + 2.397895 = 3.091042
3.091042 = 3.091042

ln (-5 -6) + ln(-5 + 3) = ln 22
ln (-11) + ln(-2) = ln 22
But logrythms are not defined for negative numbers, so x=-5 is rejected.

Final answer: x = 8

Answer 3

Hey there!

Here's the answer.

ln(x-6)+ln(x+3)=ln(22) --> Write the problem.
ln((x-6)(x+3))=ln(22) --> Use the property i.e. ln(m)+ln(n)=ln(mn).
(x-6)(x+3)=22 --> Use one to one property of logarithms i.e. if ln(a)=ln(b), then a=b.
x^2-3x-18=22 --> Multiply x-6 and x+3.
x^2-3x-40=0 --> Write the problem.
(x-8)(x+5)=0 --> Factor out the left side of the equation.
x-8=0 or x+5=0 --> Use the zero product property i.e. if pq=0, then p=0 or q=0.
x=8 or x=-5 --> Solve each equation for x.
x=8 Since -5 is less than the restricted value, i.e. since -5 is less than 6, it is an extraneous solution.

So the answer is x=8.

Hope it help!

Answer 4

ln [(x - 6)(x + 3)] = ln(22)
(x - 6)(x + 3) = 22
x^2 - 3x - 18 - 22 = 0
x^2 - 3x - 40 = 0
x^2 - 8x + 5x - 40 = 0
(x - 8)(x + 5) = 0

x = 8
x = -5(Not possible as Ln cannot take a -ve value)

So, x = 8 is the solution.