To simplify the expression [tex]\(\log 0.01\)[/tex], follow these steps:
1. Express [tex]\(0.01\)[/tex] as a power of 10:
Notice that [tex]\(0.01\)[/tex] can be written as a fraction:
[tex]\[
0.01 = \frac{1}{100}
\][/tex]
Furthermore, [tex]\(100\)[/tex] is equal to [tex]\(10^2\)[/tex], so:
[tex]\[
\frac{1}{100} = 10^{-2}
\][/tex]
Therefore:
[tex]\[
0.01 = 10^{-2}
\][/tex]
2. Apply the logarithm property for powers:
We use the property of logarithms that states [tex]\(\log(a^b) = b \log(a)\)[/tex]. In this case, we have:
[tex]\[
\log(10^{-2})
\][/tex]
Applying the property:
[tex]\[
\log(10^{-2}) = -2 \log(10)
\][/tex]
3. Simplify using the known value:
We know that the logarithm base 10 of 10 is 1:
[tex]\[
\log(10) = 1
\][/tex]
Therefore:
[tex]\[
-2 \log(10) = -2 \cdot 1 = -2
\][/tex]
So, the simplified expression is:
[tex]\[
\log 0.01 = -2
\][/tex]
