Question

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

Answer 1

To solve the equation \( \ln(10 - 3x) = \ln(-4x) \), we start by using the property of logarithms that states if \( \ln(a) = \ln(b) \), then \( a = b \), provided \( a > 0 \) and \( b > 0 \).

From the equation, we can set the arguments of the logarithms equal to each other:

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

Next, we rearrange the equation to isolate \( x \):

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

However, we need to check if this solution gives positive values for the arguments of the logarithms. We will substitute \( x = -10 \) back into the arguments:

1. For the left side:
\[
10 - 3(-10) = 10 + 30 = 40
\]

2. For the right side:
\[
-4(-10) = 40
\]

Both arguments are positive, so \( \ln(40) = \ln(40) \) holds, and thus the solution is valid.

Therefore, the solution to the equation \( \ln(10 - 3x) = \ln(-4x) \) is:

\[
\boxed{-10}
\]