Question

If \(\log_8 x = \frac{1}{3}\), find the value of \(x\).

• A. 2
• B. 4
• C. 8
• D. 1

Hint:

Remember that a fractional exponent like \(1/n\) is just another way of writing the \(n^{th}\) root. So, \(x^{1/2}\) is the square root, and \(x^{1/3}\) is the cube root.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Logarithms are the inverse of exponential functions. The statement \(\log_b a = c\) is mathematically equivalent to the exponential form \(b^c = a\).

Step 2: Key Formula or Approach
Convert the logarithmic equation to its exponential form: \[ \log_b x = y \implies x = b^y \]

Step 3: Detailed Explanation
Given the equation: \[ \log_8 x = \frac{1}{3} \] 1. Identify the base (\(b = 8\)) and the exponent (\(y = 1/3\)). 2. Rewrite in exponential form: \[ x = 8^{1/3} \] 3. Calculate the cube root of 8: \[ x = \sqrt[3]{8} = \sqrt[3]{2 \times 2 \times 2} = 2 \]

Step 4: Final Answer
The value of \(x\) is 2.