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 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.
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Answers
GPT-4o mini
To analyze the logarithmic functions \( f(x) = \log_3 x \) and \( g(x) = \log_7 x \), we can consider their key features.
1. **Domain**: Both functions are defined for \( x > 0 \).
2. **Vertical Asymptote**: Both functions approach negative infinity as \( x \) approaches 0 from the right. Therefore, they do not have different vertical asymptotes.
3. **Shape**: Both functions are increasing and have a similar shape.
4. **X-intercepts**: Both functions cross the x-axis at \( x = 1 \), so they have the same x-intercept.
5. **Rate of Increase**: The base of the logarithm determines the rate of increase. Since the base for \( f \) (which is 3) is smaller than the base for \( g \) (which is 7), \( f(x) \) will increase at a faster rate than \( g(x) \).
Based on this analysis, the correct statement 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**: Both functions approach negative infinity as \( x \) approaches 0 from the right. Therefore, they do not have different vertical asymptotes.
3. **Shape**: Both functions are increasing and have a similar shape.
4. **X-intercepts**: Both functions cross the x-axis at \( x = 1 \), so they have the same x-intercept.
5. **Rate of Increase**: The base of the logarithm determines the rate of increase. Since the base for \( f \) (which is 3) is smaller than the base for \( g \) (which is 7), \( f(x) \) will increase at a faster rate than \( g(x) \).
Based on this analysis, the correct statement is:
**"Functions \( f(x) \) and \( g(x) \) will have similar shapes but will increase at different rates."**