Find the value of :
` log _ 2 ,0.125`

Answer

To find the value of \( \log_2(0.125) \), we can follow these steps:

### Step 1: Rewrite 0.125 as a fraction
0.125 can be expressed as a fraction. Since there are three decimal places, we can write:
\[ 0.125 = \frac{125}{1000} \]

### Step 2: Simplify the fraction
Next, we can simplify \( \frac{125}{1000} \):
\[ \frac{125}{1000} = \frac{1}{8} \]

### Step 3: Express 1/8 as a power of 2
Now, we can express \( \frac{1}{8} \) as a power of 2. Since \( 8 = 2^3 \), we have:
\[ \frac{1}{8} = \frac{1}{2^3} = 2^{-3} \]

### Step 4: Substitute back into the logarithm
Now we can substitute \( 0.125 \) with \( 2^{-3} \) in the logarithm:
\[ \log_2(0.125) = \log_2(2^{-3}) \]

### Step 5: Apply the logarithmic identity
Using the property of logarithms that states \( \log_b(a^c) = c \cdot \log_b(a) \), we can simplify:
\[ \log_2(2^{-3}) = -3 \cdot \log_2(2) \]

### Step 6: Evaluate \( \log_2(2) \)
Since \( \log_2(2) = 1 \), we can substitute this value:
\[ -3 \cdot \log_2(2) = -3 \cdot 1 = -3 \]

### Final Answer
Thus, the value of \( \log_2(0.125) \) is:
\[ \log_2(0.125) = -3 \]

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