Question
Given the graph of a logarithmic function above, a student said since the point (9,2)
is on the graph and exponential functions and logarithmic functions are inverses of each other, then the point (−9,−2)
would be on the graph of an exponential function.
Is the student's reasoning and answer correct? If so, how do you know? If not, explain the student's error and correct his reasoning.
(3 points)
is on the graph and exponential functions and logarithmic functions are inverses of each other, then the point (−9,−2)
would be on the graph of an exponential function.
Is the student's reasoning and answer correct? If so, how do you know? If not, explain the student's error and correct his reasoning.
(3 points)
Answers
Answered by
GPT-4o mini
The student's reasoning is incorrect, and here's why:
1. **Understanding Inverses**: The student correctly states that logarithmic and exponential functions are inverses of each other. This means that if a point \((a, b)\) lies on the graph of a logarithmic function \(y = \log_b(x)\), then the point \((b, a)\) will lie on the graph of the corresponding exponential function \(x = b^y\).
2. **Analyzing the Given Point**: The point \((9, 2)\) implies that for the logarithmic function, we have:
\[
\log_b(9) = 2
\]
This means that \(b^2 = 9\) (where \(b\) is the base of the logarithm). Thus:
\[
b = 3 \quad (\text{since } 3^2 = 9) \text{ or } b = -3 \text{ (but } b \text{ must be positive for logarithms)}
\]
3. **Finding the Corresponding Exponential Point**: The inverse relationship implies that the corresponding point on the exponential function \(y = b^x\) would be:
\[
(2, 9) \text{ (not (−9,−2))}
\]
4. **Examining the Student’s Assertion**: The student incorrectly claims that the point \((-9, -2)\) would be on the graph of an exponential function. Instead, the correct corresponding point, based on the logarithmic point \((9, 2)\), is \((2, 9)\).
In summary, the correct point that corresponds to the logarithmic point \((9, 2)\) on an exponential graph is \((2, 9)\). The student's assertion of \((-9, -2)\) is incorrect, as the signs are not related in the context of logarithmic and exponential functions. The reasoning error stems from misunderstanding the nature of inverse relationships; the coordinates don't simply change signs when moving between logarithmic and exponential functions.
1. **Understanding Inverses**: The student correctly states that logarithmic and exponential functions are inverses of each other. This means that if a point \((a, b)\) lies on the graph of a logarithmic function \(y = \log_b(x)\), then the point \((b, a)\) will lie on the graph of the corresponding exponential function \(x = b^y\).
2. **Analyzing the Given Point**: The point \((9, 2)\) implies that for the logarithmic function, we have:
\[
\log_b(9) = 2
\]
This means that \(b^2 = 9\) (where \(b\) is the base of the logarithm). Thus:
\[
b = 3 \quad (\text{since } 3^2 = 9) \text{ or } b = -3 \text{ (but } b \text{ must be positive for logarithms)}
\]
3. **Finding the Corresponding Exponential Point**: The inverse relationship implies that the corresponding point on the exponential function \(y = b^x\) would be:
\[
(2, 9) \text{ (not (−9,−2))}
\]
4. **Examining the Student’s Assertion**: The student incorrectly claims that the point \((-9, -2)\) would be on the graph of an exponential function. Instead, the correct corresponding point, based on the logarithmic point \((9, 2)\), is \((2, 9)\).
In summary, the correct point that corresponds to the logarithmic point \((9, 2)\) on an exponential graph is \((2, 9)\). The student's assertion of \((-9, -2)\) is incorrect, as the signs are not related in the context of logarithmic and exponential functions. The reasoning error stems from misunderstanding the nature of inverse relationships; the coordinates don't simply change signs when moving between logarithmic and exponential functions.