If \(\log_8 x = \frac{1}{3}\), find the value of \(x\).
Show 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.
Step 1: Understanding the Concept:
The question requires converting a logarithmic equation into its equivalent exponential form. Step 2: Key Formula or Approach:
The fundamental definition of a logarithm states that if $\log_b y = a$, then it can be rewritten as $y = b^a$. Step 3: Detailed Explanation:
Using the definition:
\[ \log_8 x = \frac{1}{3} \implies x = 8^{\frac{1}{3}} \]
We need to calculate the cube root of 8. Express 8 as a power of 2:
\[ 8 = 2^3 \]
Substitute this back into the equation:
\[ x = \left(2^3\right)^{\frac{1}{3}} \]
Using the exponent rule $(a^m)^n = a^{m \times n}$:
\[ x = 2^{3 \times \frac{1}{3}} = 2^1 = 2 \] Step 4: Final Answer:
The value of $x$ is 2.
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