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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.
### 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.
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