Which of the following correctly solves the equation log3(2x−1)=log3(3x)

Question

?(1 point)
Responses

x=−1
x equals negative 1

x=1
x equals 1

no solution
no solution

x=15

Answer 1

To solve the equation \( \log_3(2x - 1) = \log_3(3x) \), we can use the property that if \( \log_a(b) = \log_a(c) \), then \( b = c \).

Applying this property, we set the arguments equal to each other:

\[ 2x - 1 = 3x \]

Now, we can solve for \( x \):

1. Subtract \( 2x \) from both sides:

\[ -1 = 3x - 2x \]

2. This simplifies to:

\[ -1 = x \]

Thus, the solution is:

\[ x = -1 \]

Now we need to check if this solution is valid. We need to ensure that the arguments of the logarithms are positive since logarithms are only defined for positive values:

• For \( x = -1 \):
  • \( 2(-1) - 1 = -2 - 1 = -3 \) (not valid)
  • \( 3(-1) = -3 \) (not valid)

Since the arguments of the logarithms are not valid (they can't be negative), the solution \( x = -1 \) does not work.

Therefore, there is no solution for the equation \( \log_3(2x - 1) = \log_3(3x) \).

The correct response is:

no solution.