Solve for  
x
  in the equation below. If there are multiple answers, separate them by commas. Enter exact values as your answer.
3
ln
(
6
x
−
5
)
+
4
=
−
9

To solve the equation \( 3\ln(6x - 5) + 4 = -9 \), let's go through it step-by-step.

1. **Isolate the logarithmic term:**

   \[
   3\ln(6x - 5) = -9 - 4
   \]

   \[
   3\ln(6x - 5) = -13
   \]

2. **Divide both sides by 3 to further isolate the logarithm:**

   \[
   \ln(6x - 5) = \frac{-13}{3}
   \]

3. **Rewrite the equation in exponential form:**

   Recall that \(\ln(a) = b\) is equivalent to \(e^b = a\).

   \[
   6x - 5 = e^{\frac{-13}{3}}
   \]

4. **Solve for \( x \):**

   \[
   6x = e^{\frac{-13}{3}} + 5
   \]

   \[
   x = \frac{e^{\frac{-13}{3}} + 5}{6}
   \]

Thus, the exact solution is:

\[
x = \frac{e^{\frac{-13}{3}} + 5}{6}
\]

This is the only solution as the logarithm function has a single defined value for \( e^{\frac{-13}{3}} \).

Would you like more details or have any questions?

### Relative Questions:
1. How do you convert between logarithmic and exponential forms?
2. What is the value of \( e^{\frac{-13}{3}} \) approximately?
3. How do you solve logarithmic equations with different bases?
4. What are the restrictions on the domain of \( \ln(6x - 5) \)?
5. Can logarithmic equations have multiple solutions?

**Tip:** Always check if the argument of the logarithm is positive for all values of \( x \) in the solution set.

Solve for x in the equation below. If there are multiple answers, separate them by commas. Enter exact values as your answer.
3ln(6x − 5) + 4 = −9

Learn how to solve the logarithmic equation 3ln(6x − 5) + 4 = −9 step-by-step. This problem involves isolating the logarithmic term, converting to exponential form, and solving for x. Suitable for Grades 10-12.

Math Problem Statement

Solve for
x in the equation below. If there are multiple answers, separate them by commas. Enter exact values as your answer. 3 ln ( 6 x − 5 ) + 4

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