To solve the logarithmic equation [tex]\(\log_x 216 = 3\)[/tex], let's follow the steps outlined in the hint. We'll convert the logarithmic form to an exponential form.
1. Start with the given equation:
[tex]\[
\log_x 216 = 3
\][/tex]
2. Convert the logarithmic equation to its exponential form:
[tex]\[
x^3 = 216
\][/tex]
This step is based on the property of logarithms which states that [tex]\(\log_b (y) = x \Leftrightarrow b^x = y\)[/tex].
3. Solve the exponential equation:
To find [tex]\(x\)[/tex], we need to take the cube root of both sides of the equation:
[tex]\[
x = \sqrt[3]{216}
\][/tex]
4. Evaluate the cube root of 216:
[tex]\[
\sqrt[3]{216} \approx 5.999999999999999
\][/tex]
Given the choices:
- A) [tex]\(x = 6\)[/tex]
- B) [tex]\(x = 72\)[/tex]
- C) [tex]\(x = 36\)[/tex]
Given that [tex]\(\sqrt[3]{216}\)[/tex] is very close to 6, it means the value of [tex]\(x\)[/tex] is approximately 6. Therefore, the correct answer is:
[tex]\[
\boxed{x = 6}
\][/tex]
