Question
Solve the logarithmic equation ln(3x - 11) = ln(4 + 2x).
Answers
GPT-4o mini
To solve the logarithmic equation \( \ln(3x - 11) = \ln(4 + 2x) \), we can utilize the property that if \( \ln(a) = \ln(b) \), then \( a = b \). Therefore, we can set the arguments of the logarithms equal to each other:
\[
3x - 11 = 4 + 2x
\]
Next, we will solve for \( x \) by isolating it. First, we can subtract \( 2x \) from both sides:
\[
3x - 2x - 11 = 4
\]
This simplifies to:
\[
x - 11 = 4
\]
Now, add 11 to both sides:
\[
x = 15
\]
Now we need to verify if this solution is valid. We need to check that both sides are defined and positive. We first evaluate \( 3x - 11 \) and \( 4 + 2x \) at \( x = 15 \):
1. Calculate \( 3x - 11 \):
\[
3(15) - 11 = 45 - 11 = 34
\]
which is positive.
2. Calculate \( 4 + 2x \):
\[
4 + 2(15) = 4 + 30 = 34
\]
which is also positive.
Since both expressions are positive, the solution \( x = 15 \) is valid.
Thus, the solution to the equation \( \ln(3x - 11) = \ln(4 + 2x) \) is:
\[
\boxed{15}
\]
\[
3x - 11 = 4 + 2x
\]
Next, we will solve for \( x \) by isolating it. First, we can subtract \( 2x \) from both sides:
\[
3x - 2x - 11 = 4
\]
This simplifies to:
\[
x - 11 = 4
\]
Now, add 11 to both sides:
\[
x = 15
\]
Now we need to verify if this solution is valid. We need to check that both sides are defined and positive. We first evaluate \( 3x - 11 \) and \( 4 + 2x \) at \( x = 15 \):
1. Calculate \( 3x - 11 \):
\[
3(15) - 11 = 45 - 11 = 34
\]
which is positive.
2. Calculate \( 4 + 2x \):
\[
4 + 2(15) = 4 + 30 = 34
\]
which is also positive.
Since both expressions are positive, the solution \( x = 15 \) is valid.
Thus, the solution to the equation \( \ln(3x - 11) = \ln(4 + 2x) \) is:
\[
\boxed{15}
\]