Questions: Solve the logarithmic equation. Express all solutions in exact form. log4(x^3+8) = 2 Select the correct choice below and fill in any answer boxes in your choice. A. The solution set is . (Simplify your answer. Use a comma to separate answers as needed.) B. The solution is the empty set.

Solve the logarithmic equation. Express all solutions in exact form.
log4(x^3+8) = 2

Select the correct choice below and fill in any answer boxes in your choice.
A. The solution set is .
(Simplify your answer. Use a comma to separate answers as needed.)
B. The solution is the empty set.

Transcript text: Solve the logarithmic equation. Express all solutions in exact form. \[ \log _{4}\left(x^{3}+8\right)=2 \] Select the correct choice below and fill in any answer boxes in your choice. A. The solution set is $\{$ ). $\square$ (Simplify your answer. Use a comma to separate answers as needed.) B. The solution is the empty set.

Solution

Solution Steps

To solve the logarithmic equation $\log_{4}(x^{3}+8)=2$, we need to convert the logarithmic equation into an exponential form. This involves rewriting the equation as $x^{3} + 8 = 4^{2}$. Then, solve for $x$ by isolating $x^{3}$ and taking the cube root of both sides.

Step 1: Convert the Logarithmic Equation

We start with the equation: \[ \log_{4}(x^{3}+8) = 2 \] To convert this logarithmic equation into exponential form, we rewrite it as: \[ x^{3} + 8 = 4^{2} \]

Step 2: Simplify the Exponential Equation

Calculating $4^{2}$ gives us: \[ x^{3} + 8 = 16 \] Next, we isolate $x^{3}$: \[ x^{3} = 16 - 8 \] This simplifies to: \[ x^{3} = 8 \]

Step 3: Solve for $x$

Taking the cube root of both sides, we find: \[ x = \sqrt[3]{8} \] This simplifies to: \[ x = 2 \]

Step 4: Consider Complex Solutions

The complete solution set also includes complex roots. The cubic equation $x^{3} - 8 = 0$ has the following solutions: \[ x = 2, \quad x = -1 - \sqrt{3}i, \quad x = -1 + \sqrt{3}i \]