Question

Solve the logarithmic equation ln(10−3x)=ln(−4x)
.(1 point)
x=

Answer 1

To solve the equation \( \ln(10 - 3x) = \ln(-4x) \), we can apply the property of logarithms that states if \( \ln(a) = \ln(b) \), then \( a = b \) (as long as both \( a \) and \( b \) are positive).

Thus, we equate the arguments of the logarithms:

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

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

1. Add \( 3x \) to both sides:
\[
10 = -4x + 3x
\]
\[
10 = -x
\]

2. Now multiply both sides by -1:
\[
x = -10
\]

Next, we need to check if this solution satisfies the conditions necessary for the logarithms to be defined. Specifically, both \( 10 - 3x \) and \( -4x \) must be positive:

1. Check \( 10 - 3x = 10 - 3(-10) = 10 + 30 = 40 \) (which is positive).
2. Check \( -4x = -4(-10) = 40 \) (which is also positive).

Since both expressions are positive, the solution is valid.

Thus, the final answer is:

\[
\boxed{-10}
\]