The value of `"log"_(8) 128`, is

Options

A. `(7)/(3)`
B. 16
C. `(3)/(7)`
D. `(1)/(16)`

Explanation

To find the value of \( \log_{8} 128 \), we can follow these steps:

### Step 1: Express the numbers in terms of powers of 2
We know that:
- \( 8 = 2^3 \)
- \( 128 = 2^7 \)

### Step 2: Rewrite the logarithm using the change of base formula
Using the property of logarithms that states \( \log_{a^m} b^n = \frac{n}{m} \log_{b} a \), we can rewrite \( \log_{8} 128 \) as:
\[ \log_{8} 128 = \log_{2^3} 2^7 \]

### Step 3: Apply the property of logarithms
Using the property mentioned:
\[ \log_{2^3} 2^7 = \frac{7}{3} \log_{2} 2 \]

### Step 4: Simplify using the fact that \( \log_{2} 2 = 1 \)
Since \( \log_{2} 2 = 1 \):
\[ \log_{2^3} 2^7 = \frac{7}{3} \cdot 1 = \frac{7}{3} \]

### Final Answer
Thus, the value of \( \log_{8} 128 \) is:
\[ \frac{7}{3} \]

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