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 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.
Answers
GPT-4o mini
To compare the functions \( f(x) = \log_3(x) \) and \( g(x) = \log_7(x) \):
1. **Similar shapes**: Both functions are logarithmic and will exhibit a similar shape. They both increase gradually as \( x \) increases, and they both approach negative infinity as \( x \) approaches 0 from the right.
2. **Different rates of increase**: While both functions are increasing, the base of the logarithm affects the rate at which each function increases. The function \( f(x) \) increases more rapidly for smaller values of \( x \) compared to \( g(x) \) because \( \log_3(x) > \log_7(x) \) for all \( x > 1 \). Thus, the two functions increase at different rates.
3. **Vertical asymptotes**: Neither function has a vertical asymptote in the conventional sense because both functions are defined for \( x > 0 \) and approach negative infinity as \( x \) approaches 0. They do not have vertical asymptotes at any point along the x-axis.
4. **X-intercepts**: Both functions have the same x-intercept at \( x = 1 \) since \( \log_b(1) = 0 \) for any base \( b \). Therefore, they do not have different x-intercepts.
5. **Horizontal asymptotes**: Logarithmic functions do not have horizontal asymptotes because they continue to increase without bound as \( x \) tends to infinity.
Based on these points, the correct statement that describes the similarities and differences between \( f(x) \) and \( g(x) \) is:
**Functions \( f(x) \) and \( g(x) \) will have similar shapes but will increase at different rates.**
1. **Similar shapes**: Both functions are logarithmic and will exhibit a similar shape. They both increase gradually as \( x \) increases, and they both approach negative infinity as \( x \) approaches 0 from the right.
2. **Different rates of increase**: While both functions are increasing, the base of the logarithm affects the rate at which each function increases. The function \( f(x) \) increases more rapidly for smaller values of \( x \) compared to \( g(x) \) because \( \log_3(x) > \log_7(x) \) for all \( x > 1 \). Thus, the two functions increase at different rates.
3. **Vertical asymptotes**: Neither function has a vertical asymptote in the conventional sense because both functions are defined for \( x > 0 \) and approach negative infinity as \( x \) approaches 0. They do not have vertical asymptotes at any point along the x-axis.
4. **X-intercepts**: Both functions have the same x-intercept at \( x = 1 \) since \( \log_b(1) = 0 \) for any base \( b \). Therefore, they do not have different x-intercepts.
5. **Horizontal asymptotes**: Logarithmic functions do not have horizontal asymptotes because they continue to increase without bound as \( x \) tends to infinity.
Based on these points, the correct statement that describes the similarities and differences between \( f(x) \) and \( g(x) \) is:
**Functions \( f(x) \) and \( g(x) \) will have similar shapes but will increase at different rates.**