The value of `log_(4)` 128 is

Options

A. `(5)/(3)`
B. 4
C. `(7)/(2)`
D. `(9)/(2)`

Answer

To find the value of \( \log_{4} 128 \), we can follow these steps:

### Step 1: Rewrite the logarithm in terms of base 2
First, we can express both 4 and 128 as powers of 2:
- \( 4 = 2^2 \)
- \( 128 = 2^7 \)

So, we can rewrite the logarithm:
\[ \log_{4} 128 = \log_{2^2} 2^7 \]

### Step 2: Use the change of base formula
Using the change of base formula for logarithms, we have:
\[ \log_{a^b} c^d = \frac{d}{b} \log_{a} c \]
Applying this to our logarithm:
\[ \log_{2^2} 2^7 = \frac{7}{2} \log_{2} 2 \]

### Step 3: Simplify using the property of logarithms
We know that \( \log_{2} 2 = 1 \) (since any logarithm of a number to its own base is 1). Therefore:
\[ \frac{7}{2} \log_{2} 2 = \frac{7}{2} \cdot 1 = \frac{7}{2} \]

### Final Answer
Thus, the value of \( \log_{4} 128 \) is:
\[ \log_{4} 128 = \frac{7}{2} \]

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