Let's solve the given equation step by step.

Given Equation:

$\log_2(x^2 - x - 4) = 2$

Step 1: Convert the logarithmic equation to an exponential form.

The logarithmic equation can be rewritten in exponential form using the property: $\log_b(A) = C \quad \Rightarrow \quad A = b^C$ In this case: $x^2 - x - 4 = 2^2 = 4$

Step 2: Solve the quadratic equation.

Now we have the quadratic equation: $x^2 - x - 4 = 4$ Simplifying: $x^2 - x - 4 - 4 = 0$ $x^2 - x - 8 = 0$

Step 3: Solve the quadratic equation using the quadratic formula.

The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For the equation $x^2 - x - 8 = 0$, the coefficients are:

• $a = 1$
• $b = -1$
• $c = -8$

Substituting these values into the quadratic formula: $x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-8)}}{2(1)}$ $x = \frac{1 \pm \sqrt{1 + 32}}{2}$ $x = \frac{1 \pm \sqrt{33}}{2}$

Step 4: Express the exact values.

The two exact solutions are: $x = \frac{1 + \sqrt{33}}{2}, \quad x = \frac{1 - \sqrt{33}}{2}$

Step 5: Find the decimal approximations.

Now we calculate the decimal approximations.

Let's proceed with these calculations.### Exact Values: $x = \frac{1 + \sqrt{33}}{2}, \quad x = \frac{1 - \sqrt{33}}{2}$ These correspond to: $x = 3.372, \quad x = -2.372$

Decimal Approximations:

$x \approx 3.372, \quad x \approx -2.372$

However, since logarithmic functions are only defined for positive values of their arguments, we must check if both solutions are valid. Substituting $x = -2.372$ into $x^2 - x - 4$ yields a negative result, which is not valid for the logarithmic function.

Final Solution:

• Exact value: $x = \frac{1 + \sqrt{33}}{2}$
• Decimal approximation: $x \approx 3.372$

Would you like more details on this or have any questions?

Here are some related questions:

1. How do you solve logarithmic equations in general?
2. What are the properties of logarithms?
3. How can you verify the domain of logarithmic functions?
4. Can all quadratic equations be solved using the quadratic formula?
5. What are other methods to solve quadratic equations?

Tip: Always verify the domain of logarithmic functions to exclude invalid solutions.