To determine which table represents the graph of a logarithmic function in the form [tex]\( y = \log_b(x) \)[/tex] with [tex]\( b > 1 \)[/tex], we need to check if the [tex]\( y \)[/tex]-values given in the table correspond to the logarithmic representation of the [tex]\( x \)[/tex]-values.
Let's examine the first table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
\frac{1}{8} & -3 \\
\hline
\frac{1}{4} & -2 \\
\hline
\frac{1}{2} & -1 \\
\hline
1 & 0 \\
\hline
2 & 1 \\
\hline
\end{array}
\][/tex]
For a logarithmic function [tex]\( y = \log_b(x) \)[/tex] with [tex]\( b > 1 \)[/tex], we need to check if the [tex]\( y \)[/tex]-values fit the equation when [tex]\( b = 2 \)[/tex]. Recall that [tex]\( \log_b(x) \)[/tex] represents the power to which [tex]\( b \)[/tex] must be raised to get [tex]\( x \)[/tex]. So, for base 2:
- [tex]\( \log_2\left(\frac{1}{8}\right) = -3 \)[/tex]
- [tex]\( \log_2\left(\frac{1}{4}\right) = -2 \)[/tex]
- [tex]\( \log_2\left(\frac{1}{2}\right) = -1 \)[/tex]
- [tex]\( \log_2(1) = 0 \)[/tex]
- [tex]\( \log_2(2) = 1 \)[/tex]
Let's interpret each pair:
1. [tex]\( x = \frac{1}{8} \rightarrow y = -3 \)[/tex]. This means [tex]\( 2^{-3} = \frac{1}{8} \)[/tex], which is correct.
2. [tex]\( x = \frac{1}{4} \rightarrow y = -2 \)[/tex]. This means [tex]\( 2^{-2} = \frac{1}{4} \)[/tex], which is correct.
3. [tex]\( x = \frac{1}{2} \rightarrow y = -1 \)[/tex]. This means [tex]\( 2^{-1} = \frac{1}{2} \)[/tex], which is correct.
4. [tex]\( x = 1 \rightarrow y = 0 \)[/tex]. This means [tex]\( 2^0 = 1 \)[/tex], which is correct.
5. [tex]\( x = 2 \rightarrow y = 1 \)[/tex]. This means [tex]\( 2^1 = 2 \)[/tex], which is correct.
Since all the [tex]\( y \)[/tex]-values match the expected logarithmic values with base 2, the first table represents the graph of a logarithmic function.
Thus, the table given below represents the graph [tex]\( y = \log_b(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
\frac{1}{8} & -3 \\
\hline
\frac{1}{4} & -2 \\
\hline
\frac{1}{2} & -1 \\
\hline
1 & 0 \\
\hline
2 & 1 \\
\hline
\end{array}
\][/tex]
The result confirms that this table corresponds to a logarithmic function when [tex]\( b > 1 \)[/tex], specifically with [tex]\( b = 2 \)[/tex].
