Solve the equation.
-ln (45 + 6x) = 2 ln (x + 3)
A) -6
B) 6, -6
C) 6

Question

Solve the equation.
-ln (45 + 6x) = 2 ln (x + 3)
A) -6 
B) 6, -6 
C) 6

Quizplus · Accepted Answer

We can start by simplifying the equation using the properties of logarithms:

ln (45 + 6x) = ln((x + 3)^2)

Now we can eliminate the natural logarithm and solve for x:

45 + 6x = (x + 3)^2

45 + 6x = x^2 + 6x + 9

x^2 - 36 = 0

(x - 6)(x + 6) = 0

x = 6, -6

But we must check for extraneous solutions by substitute each value into the original equation and make sure the arguments of the natural logarithms are positive:

For x = 6:

ln(45 + 6(6)) = 2ln(6 + 3)

ln(81) = 2ln(9)

4.3944 = 4.3944

The solution x=6 satisfies the original equation.

For x = -6:

ln(45 + 6(-6)) = 2ln(-6 + 3)

ln(9) = 2ln(-3)

2.1972 ≠ NaN

The solution x=-6 is extraneous since the argument of the natural logarithm is negative.

Therefore, the only solution to the equation is x=6, and the answer is C. We can simplify the given equation using the properties of logarithms as follows:

ln (45 + 6x) = 2 ln (x + 3)
ln (45 + 6x) = ln (x + 3)^2
45 + 6x = (x + 3)^2
45 + 6x = x^2 + 6x + 9
x^2 - 36 = 0
(x - 6)(x + 6) = 0

Therefore, the solutions are x = 6 and x = -6. However, we need to check if these solutions satisfy the original equation or not, since taking the logarithm of negative numbers is not possible.

Checking x = -6:
ln (45 - 36) = 2 ln (-6 + 3)
ln 9 = 2 ln (-3)
ln 9 = ln (-3)^2
ln 9 = ln 9
This leads to a contradiction, since ln 9 is not equal to ln (-3)^2. Therefore, x = -6 is not a valid solution.

Checking x = 6:
ln (45 + 6(6)) = 2 ln (6 + 3)
ln 81 = 2 ln 9
ln 81 = ln 9^2
ln 81 = ln 81
This is true, and hence x = 6 is the only valid solution.

Therefore, the answer is C (6).

Solve the Equation. -Ln (45 + 6x) = 2 Ln (X + 3)