Sure, I will guide you through the steps needed to find [tex]\(\log_3 x\)[/tex] for each given [tex]\(x\)[/tex] value. Here’s a step-by-step solution:
1. Determine [tex]\(\log_3 (\frac{1}{9})\)[/tex]:
- We know that [tex]\(\frac{1}{9}\)[/tex] can be written as [tex]\(3^{-2}\)[/tex] since [tex]\(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\)[/tex].
- Using the property of logarithms, [tex]\(\log_b (a^c) = c \log_b a\)[/tex], we get:
[tex]\[
\log_3 \left(\frac{1}{9}\right) = \log_3 \left(3^{-2}\right) = -2 \log_3 3
\][/tex]
- Since [tex]\(\log_3 3 = 1\)[/tex], it simplifies to:
[tex]\[
-2 \times 1 = -2
\][/tex]
Therefore, [tex]\(\log_3 \left(\frac{1}{9}\right) = -2.0\)[/tex].
2. Determine [tex]\(\log_3 (\frac{1}{3})\)[/tex]:
- We know that [tex]\(\frac{1}{3}\)[/tex] can be written as [tex]\(3^{-1}\)[/tex] since [tex]\(3^{-1} = \frac{1}{3}\)[/tex].
- Using the property of logarithms, [tex]\(\log_b (a^c) = c \log_b a\)[/tex], we get:
[tex]\[
\log_3 \left(\frac{1}{3}\right) = \log_3 \left(3^{-1}\right) = -1 \log_3 3
\][/tex]
- Since [tex]\(\log_3 3 = 1\)[/tex], it simplifies to:
[tex]\[
-1 \times 1 = -1
\][/tex]
Therefore, [tex]\(\log_3 \left(\frac{1}{3}\right) = -1.0\)[/tex].
3. Determine [tex]\(\log_3 (3)\)[/tex]:
- We know that [tex]\(3\)[/tex] can be written simply as [tex]\(3^1\)[/tex].
- Using the property of logarithms, [tex]\(\log_b (a^c) = c \log_b a\)[/tex], we get:
[tex]\[
\log_3 3 = \log_3 \left(3^1\right) = 1 \log_3 3
\][/tex]
- Since [tex]\(\log_3 3 = 1\)[/tex], it stays as:
[tex]\[
1 \times 1 = 1
\][/tex]
Therefore, [tex]\(\log_3 (3) = 1.0\)[/tex].
Now, let’s put these results into the table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
x & \log_3 x \\
\hline
\frac{1}{9} & -2.0 \\
\frac{1}{3} & -1.0 \\
3 & 1.0 \\
\hline
\end{tabular}
\][/tex]
These values represent the logarithms of the given [tex]\( x \)[/tex] values with base 3.
