To evaluate the logarithm \( \log(10^{22} \cdot 4) \), we can use the properties of logarithms.
The properties we need are:
- \( \log(a \cdot b) = \log a + \log b \)
- \( \log(a^b) = b \cdot \log a \)
Applying the first property to our expression:
\[ \log(10^{22} \cdot 4) = \log(10^{22}) + \log(4) \]
Now applying the second property to \( \log(10^{22}) \):
\[ \log(10^{22}) = 22 \cdot \log(10) \]
Since \( \log(10) = 1 \):
\[ \log(10^{22}) = 22 \]
Substituting back, we get:
\[ \log(10^{22} \cdot 4) = 22 + \log(4) \]
So the correct evaluation is:
\[ \log(10^{22} \cdot 4) = 22 + \log(4) \]
Among your provided responses, the second one is the correct evaluation:
\[ \log(10^{22} \cdot 4) = 22 + \log(4) \]