We are tasked with solving the equation:

$3 \ln(2x + 1) = 5$

Step 1: Isolate the logarithmic expression

First, divide both sides of the equation by 3 to isolate the natural logarithm term:

$\ln(2x + 1) = \frac{5}{3}$

Step 2: Eliminate the logarithm using exponentiation

The natural logarithm $\ln(y)$ is the inverse of the exponential function $e^y$. Thus, we exponentiate both sides of the equation:

$2x + 1 = e^{\frac{5}{3}}$

Step 3: Solve for $x$

Now, solve for $x$ by subtracting 1 from both sides:

$2x = e^{\frac{5}{3}} - 1$

Finally, divide by 2:

$x = \frac{e^{\frac{5}{3}} - 1}{2}$

Final Answer:

The exact solution is:

$x = \frac{e^{\frac{5}{3}} - 1}{2}$

Would you like further details on this solution or have any other questions?

5 Related Questions:

1. How can the natural logarithm and exponential functions be used in solving equations?
2. What are the properties of logarithms that simplify solving equations like this?
3. What are the applications of logarithmic equations in real-world problems?
4. How would the equation change if the logarithm base were different, such as $\log_{10}$?
5. How can you check if the solution to a logarithmic equation is valid?

Tip:

Always check the domain of logarithmic functions — the argument must be positive for real solutions.