Question
Compare the two logarithmic functions f(x)=log3 x
and g(x)=log7 x
. 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 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.
Functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis will have similar shapes but will have different x -intercepts.
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 vertical asymptotes.
and g(x)=log7 x
. 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 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.
Functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis will have similar shapes but will have different x -intercepts.
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 vertical asymptotes.
Answers
GPT-4o mini
The correct statement that describes the similarities and differences between the logarithmic functions \( f(x) = \log_3 x \) and \( g(x) = \log_7 x \) is:
**"Functions \( f(x) \) and \( g(x) \) will have similar shapes but will increase at different rates."**
**Explanation:**
- Both functions are logarithmic functions and share the same general shape (they both increase slowly and approach infinity as \( x \) increases).
- However, they have different bases (3 and 7), which means they will increase at different rates. Specifically, \( g(x) = \log_7 x \) will increase more slowly than \( f(x) = \log_3 x \) because the base of \( g(x) \) is larger.
- Both functions have an x-intercept at \( x = 1 \) (since \( \log_b(1) = 0 \) for any base \( b \)), meaning they do not differ in their x-intercepts.
- Neither function has horizontal or vertical asymptotes; they both approach infinity and have a vertical asymptote at \( x = 0 \).
Thus, the key differentiation lies in their rates of growth rather than their shapes, intercepts, or asymptotes.
**"Functions \( f(x) \) and \( g(x) \) will have similar shapes but will increase at different rates."**
**Explanation:**
- Both functions are logarithmic functions and share the same general shape (they both increase slowly and approach infinity as \( x \) increases).
- However, they have different bases (3 and 7), which means they will increase at different rates. Specifically, \( g(x) = \log_7 x \) will increase more slowly than \( f(x) = \log_3 x \) because the base of \( g(x) \) is larger.
- Both functions have an x-intercept at \( x = 1 \) (since \( \log_b(1) = 0 \) for any base \( b \)), meaning they do not differ in their x-intercepts.
- Neither function has horizontal or vertical asymptotes; they both approach infinity and have a vertical asymptote at \( x = 0 \).
Thus, the key differentiation lies in their rates of growth rather than their shapes, intercepts, or asymptotes.