Question
Compare the two logarithmic functions f(x)=log3x
and g(x)=log7x
. Which statement correctly describes the similarities and differences between these two functions?(1 point)
Responses
Functions f(x)
and g(x)
will have similar shapes but will have different vertical asymptotes.
Functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis will have similar shapes but will have different vertical asymptotes.
Functions f(x)
and g(x)
will have similar shapes but will increase at different rates.
Functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis will have similar shapes but will increase at different rates.
Functions f(x)
and g(x)
will have similar shapes but will have different horizontal asymptotes.
Functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis will have similar shapes but will have different horizontal asymptotes.
Functions f(x)
and g(x)
will have similar shapes but will have different x-intercepts.
Functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis will have similar shapes but will have different x -intercepts.
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and g(x)=log7x
. Which statement correctly describes the similarities and differences between these two functions?(1 point)
Responses
Functions f(x)
and g(x)
will have similar shapes but will have different vertical asymptotes.
Functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis will have similar shapes but will have different vertical asymptotes.
Functions f(x)
and g(x)
will have similar shapes but will increase at different rates.
Functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis will have similar shapes but will increase at different rates.
Functions f(x)
and g(x)
will have similar shapes but will have different horizontal asymptotes.
Functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis will have similar shapes but will have different horizontal asymptotes.
Functions f(x)
and g(x)
will have similar shapes but will have different x-intercepts.
Functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis will have similar shapes but will have different x -intercepts.
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Answers
GPT-4o mini
To compare the two logarithmic functions \( f(x) = \log_3{x} \) and \( g(x) = \log_7{x} \), let's analyze their properties.
1. **Domain**: Both functions are defined for \( x > 0 \).
2. **Vertical Asymptote**: Neither function has a vertical asymptote; they are both defined for all positive \( x \).
3. **Horizontal Asymptote**: Both functions do not have a horizontal asymptote either.
4. **X-Intercept**: Both functions have the same x-intercept, which is at \( x = 1 \), because \( \log_b{1} = 0 \) for any base \( b \).
5. **Growth Rate**: The main difference is in their growth rates. Since \( \log_3{x} \) and \( \log_7{x} \) are logarithmic functions of different bases, they will have different slopes and hence increase at different rates.
The correct statement that describes the similarities and differences between these two functions is:
- **Functions \( f(x) \) and \( g(x) \) will have similar shapes but will increase at different rates.**
1. **Domain**: Both functions are defined for \( x > 0 \).
2. **Vertical Asymptote**: Neither function has a vertical asymptote; they are both defined for all positive \( x \).
3. **Horizontal Asymptote**: Both functions do not have a horizontal asymptote either.
4. **X-Intercept**: Both functions have the same x-intercept, which is at \( x = 1 \), because \( \log_b{1} = 0 \) for any base \( b \).
5. **Growth Rate**: The main difference is in their growth rates. Since \( \log_3{x} \) and \( \log_7{x} \) are logarithmic functions of different bases, they will have different slopes and hence increase at different rates.
The correct statement that describes the similarities and differences between these two functions is:
- **Functions \( f(x) \) and \( g(x) \) will have similar shapes but will increase at different rates.**