To solve for [tex]\(\log(x)\)[/tex] for each given value of [tex]\(x\)[/tex], we need to understand what [tex]\(\log(x)\)[/tex] represents. The logarithm [tex]\(\log_{10}(x)\)[/tex], often written simply as [tex]\(\log(x)\)[/tex], refers to the exponent to which the base 10 must be raised to produce the number [tex]\(x\)[/tex].
Given the following values of [tex]\(x\)[/tex]:
- [tex]\(10^3\)[/tex]
- [tex]\(10^2\)[/tex]
- [tex]\(10^1\)[/tex]
- [tex]\(10^0\)[/tex]
- [tex]\(10^{-1}\)[/tex]
- [tex]\(10^{-2}\)[/tex]
- [tex]\(10^{-3}\)[/tex]
- [tex]\(10^{1/2}\)[/tex]
Let's find [tex]\(\log(x)\)[/tex] for each:
1. For [tex]\(x = 10^3\)[/tex]:
[tex]\[
\log(10^3) = 3
\][/tex]
2. For [tex]\(x = 10^2\)[/tex]:
[tex]\[
\log(10^2) = 2
\][/tex]
3. For [tex]\(x = 10^1\)[/tex]:
[tex]\[
\log(10^1) = 1
\][/tex]
4. For [tex]\(x = 10^0\)[/tex]:
[tex]\[
\log(10^0) = 0
\][/tex]
5. For [tex]\(x = 10^{-1}\)[/tex]:
[tex]\[
\log(10^{-1}) = -1
\][/tex]
6. For [tex]\(x = 10^{-2}\)[/tex]:
[tex]\[
\log(10^{-2}) = -2
\][/tex]
7. For [tex]\(x = 10^{-3}\)[/tex]:
[tex]\[
\log(10^{-3}) = -3
\][/tex]
8. For [tex]\(x = 10^{1/2}\)[/tex]:
[tex]\[
\log(10^{1/2}) = \frac{1}{2}
\][/tex]
So, the completed table for [tex]\(\log(x)\)[/tex] is:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
x & 10^3 & 10^2 & 10^1 & 10^0 & 10^{-1} & 10^{-2} & 10^{-3} & 10^{1 / 2} \\
\hline
\log(x) & 3 & 2 & 1 & 0 & -1 & -2 & -3 & 0.5 \\
\hline
\end{array}
\][/tex]
