To determine which table represents the graph of a logarithmic function with both an [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-intercept, we need to check each table to see if the data points satisfy the conditions for these intercepts.
### Understanding Intercepts:
1. [tex]\( x \)[/tex]-intercept: The point where the graph crosses the [tex]\( x \)[/tex]-axis. Here, [tex]\( y = 0 \)[/tex].
2. [tex]\( y \)[/tex]-intercept: The point where the graph crosses the [tex]\( y \)[/tex]-axis. Here, [tex]\( x = 0 \)[/tex].
### Checking Each Table:
#### Table 1:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
3 & \emptyset \\
\hline
4 & -15 \\
\hline
5 & 0.585 \\
\hline
6 & 1.322 \\
\hline
7 & 1.807 \\
\hline
\end{array}
\][/tex]
- For [tex]\( x \)[/tex]-intercept: None of the given [tex]\( y \)[/tex] values is 0.
- For [tex]\( y \)[/tex]-intercept: None of the given [tex]\( x \)[/tex] values is 0.
- Conclusion: Table 1 does not have both intercepts.
#### Table 2:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-1.5 & -1 \\
\hline
-0.5 & 0.585 \\
\hline
0.5 & 1.322 \\
\hline
1.5 & 1.807 \\
\hline
2.5 & 2.169 \\
\hline
\end{array}
\][/tex]
- For [tex]\( x \)[/tex]-intercept: None of the given [tex]\( y \)[/tex] values is 0.
- For [tex]\( y \)[/tex]-intercept: None of the given [tex]\( x \)[/tex] values is 0.
- Conclusion: Table 2 does not have both intercepts.
#### Table 3:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0.5 & -0.631 \\
\hline
1.5 & 0.369 \\
\hline
2.5 & 0.834 \\
\hline
3.5 & 1.146 \\
\hline
4.5 & 1.369 \\
\hline
\end{array}
\][/tex]
- For [tex]\( x \)[/tex]-intercept: None of the given [tex]\( y \)[/tex] values is 0.
- For [tex]\( y \)[/tex]-intercept: None of the given [tex]\( x \)[/tex] values is 0.
- Conclusion: Table 3 does not have both intercepts.
Based on this analysis, none of the tables provided represent a logarithmic function with both an [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-intercept.
