Solve for the exact value of x, .x.
3, natural log, left parenthesis, 3, x, plus, 9, right parenthesis, plus, 19, equals, 13
3ln(3x+9)+19=
13

Let's solve the equation step by step:

The given equation is:

\[
3 \ln(3x + 9) + 19 = 13
\]

### Step 1: Isolate the logarithmic term
First, subtract 19 from both sides of the equation to isolate the logarithmic term:

\[
3 \ln(3x + 9) = 13 - 19
\]

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

### Step 2: Divide by 3
Next, divide both sides of the equation by 3:

\[
\ln(3x + 9) = \frac{-6}{3}
\]

\[
\ln(3x + 9) = -2
\]

### Step 3: Exponentiate both sides
To remove the natural logarithm, exponentiate both sides of the equation using the base \( e \):

\[
3x + 9 = e^{-2}
\]

### Step 4: Solve for \( x \)
Now, subtract 9 from both sides:

\[
3x = e^{-2} - 9
\]

Finally, divide by 3:

\[
x = \frac{e^{-2} - 9}{3}
\]

### Final Answer:
The exact value of \( x \) is:

\[
x = \frac{e^{-2} - 9}{3}
\]

Would you like any further details or have any questions?

Here are 5 related questions for further exploration:
1. What is the numerical value of \( e^{-2} \)?
2. How would you solve a similar equation involving \( \ln(x) \) but with a different coefficient?
3. What is the inverse function of \( \ln(x) \)?
4. How can you solve an equation where the logarithm is with a different base, not \( e \)?
5. What are the applications of natural logarithms in real-world problems?

**Tip:** When solving logarithmic equations, isolating the logarithmic term first simplifies the problem and helps in solving it step by step.

Learn how to solve the equation 3ln(3x + 9) + 19 = 13 step-by-step with detailed explanations of logarithmic and exponential functions. Suitable for advanced high school students.

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