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 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 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 have different x-intercepts.
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 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 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 have different x-intercepts.
Answers
GPT-4o mini
The two logarithmic functions, \( f(x) = \log_3 x \) and \( g(x) = \log_7 x \), indeed have similar shapes because they are both logarithmic functions, but they use different bases. This difference in base affects the rate at which the functions increase.
Now, let's analyze the statements:
1. **Functions f(x) and g(x) will have similar shapes but will increase at different rates.**
- This statement is **true**. Both functions will approach infinity as \( x \) increases, but \( \log_3 x \) will increase more quickly than \( \log_7 x \) because the base of the logarithm affects the steepness of the graph.
2. **Functions f(x) and g(x) will have similar shapes but will have different vertical asymptotes.**
- This statement is **false**. Both functions have a vertical asymptote at \( x = 0 \).
3. **Functions f(x) and g(x) will have similar shapes but will have different horizontal asymptotes.**
- This statement is **false**. There are no horizontal asymptotes for logarithmic functions; they approach infinity as \( x \) increases.
4. **Functions f(x) and g(x) will have similar shapes but will have different x-intercepts.**
- This statement is **false**. Both functions cross the x-axis at the same point, \( x = 1 \) (since \( \log_3(1) = 0 \) and \( \log_7(1) = 0 \)).
In summary, 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.**
Now, let's analyze the statements:
1. **Functions f(x) and g(x) will have similar shapes but will increase at different rates.**
- This statement is **true**. Both functions will approach infinity as \( x \) increases, but \( \log_3 x \) will increase more quickly than \( \log_7 x \) because the base of the logarithm affects the steepness of the graph.
2. **Functions f(x) and g(x) will have similar shapes but will have different vertical asymptotes.**
- This statement is **false**. Both functions have a vertical asymptote at \( x = 0 \).
3. **Functions f(x) and g(x) will have similar shapes but will have different horizontal asymptotes.**
- This statement is **false**. There are no horizontal asymptotes for logarithmic functions; they approach infinity as \( x \) increases.
4. **Functions f(x) and g(x) will have similar shapes but will have different x-intercepts.**
- This statement is **false**. Both functions cross the x-axis at the same point, \( x = 1 \) (since \( \log_3(1) = 0 \) and \( \log_7(1) = 0 \)).
In summary, 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.**