To solve the logarithmic equation \( \ln(3x - 11) = \ln(4 + 2x) \), we can use the property of logarithms that states if \( \ln(a) = \ln(b) \), then \( a = b \).
So, we set the insides of the logarithms equal to each other:
\[ 3x - 11 = 4 + 2x \]
Now, we will isolate \( x \). First, subtract \( 2x \) from both sides:
\[ 3x - 2x - 11 = 4 \]
This simplifies to:
\[ x - 11 = 4 \]
Next, add 11 to both sides to solve for \( x \):
\[ x = 4 + 11 \]
\[ x = 15 \]
Now, we should check that this value does not make either logarithm undefined (the argument must be positive):
-
For \( 3x - 11 \): \[ 3(15) - 11 = 45 - 11 = 34 > 0 \]
-
For \( 4 + 2x \): \[ 4 + 2(15) = 4 + 30 = 34 > 0 \]
Both arguments are positive, so the solution is valid.
Thus, the solution to the equation is:
\[ \boxed{15} \]