Question

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

Answer 1

To solve the 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 can set the arguments of the logarithms equal to each other:

\[
3x - 11 = 4 + 2x
\]

Next, we'll solve for \( x \). First, let's isolate \( x \) by subtracting \( 2x \) from both sides:

\[
3x - 2x - 11 = 4
\]

This simplifies to:

\[
x - 11 = 4
\]

Now, add 11 to both sides:

\[
x = 15
\]

Now, we should check if \( x = 15 \) is valid by substituting it back into the arguments of the logarithms to ensure they are positive:

For \( 3x - 11 \):

\[
3(15) - 11 = 45 - 11 = 34 > 0
\]

For \( 4 + 2x \):

\[
4 + 2(15) = 4 + 30 = 34 > 0
\]

Both expressions are positive, which means our solution is valid.

Thus, the solution is:

\[
\boxed{15}
\]