The value of `log_(8) 128` is equal to

Options

A. `(7)/(3)`
B. `16`
C. `(3)/(7)`
D. none

Answer

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

### Step 1: Rewrite the logarithm in terms of base 2
We know that both 8 and 128 can be expressed as powers of 2:
- \( 8 = 2^3 \)
- \( 128 = 2^7 \)

Thus, we can rewrite the logarithm:
\[ \log_{8} 128 = \log_{2^3} 2^7 \]

### Step 2: Apply the change of base formula
Using the change of base formula for logarithms, we have:
\[ \log_{a^b} c^d = \frac{d}{b} \log_{a} c \]
In our case:
- \( a = 2 \)
- \( b = 3 \)
- \( c = 2 \)
- \( d = 7 \)

Applying the formula:
\[ \log_{2^3} 2^7 = \frac{7}{3} \log_{2} 2 \]

### Step 3: Simplify using the property of logarithms
We know that \( \log_{2} 2 = 1 \). Therefore:
\[ \frac{7}{3} \log_{2} 2 = \frac{7}{3} \cdot 1 = \frac{7}{3} \]

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

---