Question

Solve the logarithmic equation ln(3x−11)=ln(4+2x)
.(1 point)
x=

Answer 1

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):

1. For \( 3x - 11 \):
\[
3(15) - 11 = 45 - 11 = 34 > 0
\]

2. 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:

\[
\boxed{15}
\]