Find the value of :
` log_(0.2) 5`

Answer

To find the value of \( \log_{0.2} 5 \), we can follow these steps:

### Step 1: Rewrite the base
We know that \( 0.2 \) can be expressed as a fraction:
\[ 0.2 = \frac{2}{10} = \frac{1}{5} \]
Thus, we can rewrite the logarithm:
\[ \log_{0.2} 5 = \log_{\frac{1}{5}} 5 \]

### Step 2: Apply the logarithmic identity
There is a logarithmic identity that states:
\[ \log_{a} b = -\log_{\frac{1}{a}} b \]
In our case, we can use this identity:
\[ \log_{\frac{1}{5}} 5 = -\log_{5} 5 \]

### Step 3: Evaluate \( \log_{5} 5 \)
We know that:
\[ \log_{5} 5 = 1 \]
because any number to the power of 1 is itself.

### Step 4: Substitute back into the equation
Now substituting back, we have:
\[ \log_{\frac{1}{5}} 5 = -\log_{5} 5 = -1 \]

### Final Answer
Thus, the value of \( \log_{0.2} 5 \) is:
\[ \boxed{-1} \]