Find the value of `log_(0.125) 64`

Options

A. (-2)
B. 2
C. 0
D. cant determined

Answer

To find the value of \( \log_{0.125} 64 \), we can follow these steps:

### Step 1: Rewrite the logarithm in terms of base 2
First, we need to express both 0.125 and 64 as powers of 2.

- \( 0.125 = \frac{1}{8} = \frac{1}{2^3} = 2^{-3} \)
- \( 64 = 2^6 \)

So we can rewrite the logarithm:
\[ \log_{0.125} 64 = \log_{2^{-3}} 2^6 \]

### 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:
\[ \log_{2^{-3}} 2^6 = \frac{6}{-3} \log_2 2 \]

### Step 3: Simplify the logarithm
Since \( \log_2 2 = 1 \):
\[ \log_{2^{-3}} 2^6 = \frac{6}{-3} \cdot 1 = -2 \]

### Final Answer
Thus, the value of \( \log_{0.125} 64 \) is:
\[ \boxed{-2} \]

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