To solve the equation $11 - 4 \ln(x - 6) = 0$ both graphically and algebraically, let's proceed step by step.

Step 1: Rewrite the Equation

We want to isolate $\ln(x - 6)$. Start by moving the constants around:

$11 - 4 \ln(x - 6) = 0$ $-4 \ln(x - 6) = -11$ $\ln(x - 6) = \frac{11}{4}$

Step 2: Solve Algebraically

To remove the natural logarithm, exponentiate both sides of the equation using base $e$:

$x - 6 = e^{\frac{11}{4}}$

Now, calculate $e^{\frac{11}{4}}$ using a calculator:

$x - 6 \approx e^{2.75} \approx 15.642$

Thus, solving for $x$:

$x \approx 15.642 + 6 = 21.642$

So, the solution algebraically is $x \approx 21.642$.

Step 3: Solve Graphically

To solve graphically, graph the function $y = 11 - 4 \ln(x - 6)$ and look for the x-intercept (where $y = 0$).

The equation becomes: $y = 11 - 4 \ln(x - 6)$ Graph this function using a graphing utility. The x-coordinate of the point where the graph crosses the x-axis will give us the approximate solution.

Verification

• The graph confirms that the x-intercept is approximately $x \approx 21.642$, which matches the algebraic solution.

Conclusion:

The solution to the equation $11 - 4 \ln(x - 6) = 0$ is $x \approx 21.642$ to three decimal places.

Would you like more details on any step or need further clarification?

Relative Questions:

1. How can logarithmic functions be simplified for easier graphing?
2. What is the role of the natural logarithm $\ln$ in equations like this one?
3. How do we verify solutions graphically with different functions?
4. What methods can be used to solve logarithmic equations algebraically?
5. How does changing the base of a logarithm affect solving the equation?

Tip:

When solving logarithmic equations, always ensure the argument inside the logarithm is positive, as the logarithm of a negative number or zero is undefined.