Which of the following correctly solves the equation ln(10−4x)=ln(10−3x)

Question

ln
⁡
(
10
−
4
𝑥
)
=
ln
⁡
(
10
−
3
𝑥
)
?(1 point)
Responses

x=20
𝑥
=
20
x equals 20

no solution
no solution

x=207
𝑥
=
20
7
x equals Start Fraction 20 over 7 End Fraction

Answer 1

To solve the equation \( \ln(10 - 4x) = \ln(10 - 3x) \), we can use the property that if \( \ln(a) = \ln(b) \), then \( a = b \), provided that \( a > 0 \) and \( b > 0 \).

1. First, set the arguments of the logarithms equal to each other:

   \[ 10 - 4x = 10 - 3x \]

2. Next, simplify the equation:

   Subtract \( 10 \) from both sides:

   \[ -4x = -3x \]

   Now, add \( 3x \) to both sides:

   \[ -4x + 3x = 0 \]

   \[ -x = 0 \]

   Therefore, \( x = 0 \).

3. Now, we should check whether this solution is valid by substituting \( x = 0 \) back into the original logarithmic arguments:

   For \( x = 0 \): \[ \ln(10 - 4(0)) = \ln(10) \quad \text{and} \quad \ln(10 - 3(0)) = \ln(10) \]

   Both sides are equal, which confirms that the solution is valid.

So, the correct solution to the equation is:

\[ x = 0 \]

None of the provided responses (x=20, no solution, x=20/7) are correct based on the verification of \( x = 0 \).