Question

help pls : Logarithmic Equations (Algebra) - Hard
Solve the equation: log₃(x - 1) + log₃(x + 2) = 2.

Answer 1

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Answer 2

To solve the equation

\log_3(x - 1) + \log_3(x + 2) = 2,

Step 1: Combine the logarithms using the logarithmic property

Using the property , we rewrite the equation as:

\log_3((x - 1)(x + 2)) = 2.

Step 2: Exponentiate both sides

To eliminate the logarithm, rewrite the equation in exponential form:

(x - 1)(x + 2) = 3^2.

Simplify:

(x - 1)(x + 2) = 9.

Step 3: Expand and simplify

Expand the left-hand side:

x^2 + 2x - x - 2 = 9.

Simplify:

x^2 + x - 2 = 9.

x^2 + x - 11 = 0.

Step 4: Solve the quadratic equation

The quadratic equation is:

x^2 + x - 11 = 0.

Use the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},

where , , and . Substitute these values:

x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-11)}}{2(1)}.

x = \frac{-1 \pm \sqrt{1 + 44}}{2}.

x = \frac{-1 \pm \sqrt{45}}{2}.

Simplify as :

x = \frac{-1 \pm 3\sqrt{5}}{2}.

Thus, the two solutions are:

x = \frac{-1 + 3\sqrt{5}}{2} \quad \text{and} \quad x = \frac{-1 - 3\sqrt{5}}{2}.

Step 5: Check for extraneous solutions

Since and are defined only when and , we require .

For , approximate:


\frac{-1 + 3\sqrt{5}}{2} \approx \frac{-1 + 6.708}{2} = \frac{5.708}{2} \approx 2.854 \quad (\text{valid, since } x > 1).

For , approximate:


\frac{-1 - 3\sqrt{5}}{2} \approx \frac{-1 - 6.708}{2} = \frac{-7.708}{2} \approx -3.854 \quad (\text{invalid, since } x > 1).

Final Answer:

x = \frac{-1 + 3\sqrt{5}}{2}.

Answer 3

🗿

Answer 4

ok thx