Question
Compare the two logarithmic functions \(f\left(x\right)=\log_3x\) and \(g\left(x\right)=\log_7x\). 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 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 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 x-intercepts.
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 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 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 x-intercepts.
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. **Range**: The range of both functions is all real numbers, i.e., \((-\infty, \infty)\).
3. **Vertical Asymptotes**: Both functions approach negative infinity as \(x\) approaches 0 from the right, so there are no vertical asymptotes (though they both have a vertical asymptote at \(x = 0\)).
4. **Horizontal Asymptotes**: Neither function has horizontal asymptotes.
5. **x-intercept**: Both functions have an x-intercept at \(x = 1\) since \(\log_b 1 = 0\) for any base \(b > 0\).
6. **Growth Rates**: The rate at which the functions increase differs because base \(3\) will grow more slowly than base \(7\) as \(x\) increases.
Given this analysis, the statement that accurately reflects their differences and similarities 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. **Range**: The range of both functions is all real numbers, i.e., \((-\infty, \infty)\).
3. **Vertical Asymptotes**: Both functions approach negative infinity as \(x\) approaches 0 from the right, so there are no vertical asymptotes (though they both have a vertical asymptote at \(x = 0\)).
4. **Horizontal Asymptotes**: Neither function has horizontal asymptotes.
5. **x-intercept**: Both functions have an x-intercept at \(x = 1\) since \(\log_b 1 = 0\) for any base \(b > 0\).
6. **Growth Rates**: The rate at which the functions increase differs because base \(3\) will grow more slowly than base \(7\) as \(x\) increases.
Given this analysis, the statement that accurately reflects their differences and similarities is:
**Functions \(f(x)\) and \(g(x)\) will have similar shapes but will increase at different rates.**