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

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.