If ` log_(sqrt8) b = 3 1/3`, then find the value of b.

Explanation

To solve the equation \( \log_{\sqrt{8}} b = 3 \frac{1}{3} \), we will follow these steps:

### Step 1: Convert the mixed number to an improper fraction
The mixed number \( 3 \frac{1}{3} \) can be converted to an improper fraction:
\[ 3 \frac{1}{3} = \frac{10}{3} \]

### Step 2: Rewrite the logarithmic equation in exponential form
Using the property of logarithms, we can rewrite the equation:
\[ b = (\sqrt{8})^{\frac{10}{3}} \]

### Step 3: Simplify \( \sqrt{8} \)
We know that \( \sqrt{8} = 8^{1/2} \). Since \( 8 = 2^3 \), we can write:
\[ \sqrt{8} = (2^3)^{1/2} = 2^{3/2} \]

### Step 4: Substitute back into the equation
Now substitute \( \sqrt{8} \) back into the equation for \( b \):
\[ b = (2^{3/2})^{\frac{10}{3}} \]

### Step 5: Use the power of a power property
Using the property \( (a^m)^n = a^{m \cdot n} \):
\[ b = 2^{(3/2) \cdot (10/3)} \]

### Step 6: Simplify the exponent
Calculate the exponent:
\[ (3/2) \cdot (10/3) = \frac{3 \cdot 10}{2 \cdot 3} = \frac{10}{2} = 5 \]
Thus,
\[ b = 2^5 \]

### Step 7: Calculate the final value of \( b \)
Finally, calculate \( 2^5 \):
\[ b = 32 \]

### Final Answer
The value of \( b \) is \( 32 \).

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