Which of the following is a reasonable solution to the logarithmic equation

Question

ln
(
2
−
3
x
)
=
ln
(
−
5
x
)
?(1 point)
Responses

x
=
1
x is equal to 1

x
=
−
3
x is equal to negative 3

x
=
−
1
x is equal to negative 1

x
=
3

Answer 1

To solve the logarithmic equation \( \ln(2 - 3x) = \ln(-5x) \), we can start by setting the arguments of the natural logarithm equal to each other, given that the natural logarithm is a one-to-one function:

\[ 2 - 3x = -5x \]

Next, we solve for \( x \):

1. Add \( 3x \) to both sides: \[ 2 = -5x + 3x \] \[ 2 = -2x \]

2. Divide both sides by -2: \[ x = -1 \]

Now, we need to check if this solution is valid by ensuring that the arguments of the logarithms are positive:

1. Substitute \( x = -1 \) back into the arguments:
   • For \( 2 - 3(-1) \): \[ 2 + 3 = 5 \quad (\text{which is positive}) \]
   • For \( -5(-1) \): \[ 5 \quad (\text{which is also positive}) \]

Since both arguments are positive, \( x = -1 \) is a valid solution.

• Now let's check the other provided options for validity:
  • \( x = 1 \):
    • \( 2 - 3(1) = -1 \) (not valid)
    • \( -5(1) = -5 \) (not valid)
  • \( x = -3 \):
    • \( 2 - 3(-3) = 11 \) (valid)
    • \( -5(-3) = 15 \) (valid)
  • \( x = 3 \):
    • \( 2 - 3(3) = -7 \) (not valid)
    • \( -5(3) = -15 \) (not valid)

The only reasonable solution in the context of the options given is:

\( x = -1 \).