log0,2 100 - log0,2 4

To solve the expression \( \log_{0.2}(100) - \log_{0.2}(4) \), we can use the logarithm property that states:

\[ \log_a(b) - \log_a(c) = \log_a\left(\frac{b}{c}\right) \]

So, we rewrite the expression as:

\[ \log_{0.2}\left(\frac{100}{4}\right) \]

Now, calculate \( \frac{100}{4} \):

\[ \frac{100}{4} = 25 \]

Now we have:

\[ \log_{0.2}(25) \]

Next, we can express \( 25 \) as a power of \( 0.2 \). Recall that \( 0.2 = \frac{1}{5} \), and \( 25 = 5^2 \). Thus:

\[ \log_{0.2}(25) = \log_{0.2}(5^2) = 2 \cdot \log_{0.2}(5) \]

Now we need to find \( \log_{0.2}(5) \). We can use the change of base formula:

\[ \log_{0.2}(5) = \frac{\log_{10}(5)}{\log_{10}(0.2)} \]

Using \( \log_{10}(0.2) = \log_{10}\left(\frac{1}{5}\right) = -\log_{10}(5) \), we get:

\[ \log_{0.2}(5) = \frac{\log_{10}(5)}{-\log_{10}(5)} = -1 \]

Now substitute back into our expression:

\[ \log_{0.2}(25) = 2 \cdot (-1) = -2 \]

Thus, the final result is:

\[ \log_{0.2}(100) - \log_{0.2}(4) = -2 \]