To solve the equation $3\ln(x) = 6 - 2x$ for $x$, we will proceed as follows:

Step 1: Rewrite the Equation

We start with: $3\ln(x) = 6 - 2x$ Dividing both sides by 3 gives: $\ln(x) = 2 - \frac{2x}{3}$

Step 2: Define a Function and Solve Numerically

Define the function $f(x) = \ln(x) - 2 + \frac{2x}{3}$. We want to find the values of $x$ that make $f(x) = 0$.

Step 3: Solve Numerically

Since this equation is not easily solvable algebraically, we can approximate the solution numerically.

Let’s proceed to find the approximate solutions for $x$.The solution to the equation $3\ln(x) = 6 - 2x$, correct to two decimal places, is:

$x = 1.98$

Would you like further details on this solution, or do you have any additional questions?

Here are some related questions that explore similar concepts:

1. How can logarithmic equations be solved when they involve linear or polynomial expressions?
2. What are the methods to approximate solutions for transcendental equations?
3. Can this equation have more than one real solution?
4. How does changing the constant term (e.g., from 6 to another value) affect the solution?
5. How can we verify the solution to a logarithmic equation numerically?

Tip: When solving equations involving both logarithmic and polynomial terms, numerical methods like Newton’s method or the bisection method can be very effective.