Question
Compare the two logarithmic functions f(x)=log
3
x and g(x)=log
7
x. Which statement correctly describes the similarities and differences between these two functions?
A. Functions f(x) and g(x) will have similar shapes but will have different vertical asymptotes. B. Functions f(x) and g(x) will have similar shapes but will have different horizontal asymptotes. C. Functions f(x) and g(x) will have similar shapes but will increase at different rates. D. Functions f(x) and g(x) will have similar shapes but will have different x-intercepts.
3
x and g(x)=log
7
x. Which statement correctly describes the similarities and differences between these two functions?
A. Functions f(x) and g(x) will have similar shapes but will have different vertical asymptotes. B. Functions f(x) and g(x) will have similar shapes but will have different horizontal asymptotes. C. Functions f(x) and g(x) will have similar shapes but will increase at different rates. D. 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 \( g(x) = \log_7 x \), we can analyze their properties:
1. **Domain**: Both functions are defined for \( x > 0 \).
2. **Vertical Asymptote**: Both functions approach negative infinity as \( x \) approaches 0, but they do not have a vertical asymptote since their domains start from \( x = 0 \) and go to \( x = \infty \).
3. **Horizontal Asymptote**: Neither function has a horizontal asymptote because they both increase without bound as \( x \) approaches infinity.
4. **Shape**: Both functions are increasing and have a similar curve shape typical of logarithmic functions.
5. **Rate of Increase**: The base of the logarithm affects the rate of increase. \( \log_3 x \) grows faster than \( \log_7 x \) for the same values of \( x \) because the larger base (7) results in a slower growth rate compared to a smaller base (3).
6. **X-Intercepts**: Both functions have an x-intercept at \( x = 1 \) because \( \log_b(1) = 0 \) for any base \( b > 0 \).
Considering these points, the most accurate statement that describes the similarities and differences is:
**C. Functions f(x) and g(x) will have similar shapes but will increase at different rates.**
This captures that both functions have a similar shape typical of logarithmic functions, but their growth rates differ due to their respective bases.
1. **Domain**: Both functions are defined for \( x > 0 \).
2. **Vertical Asymptote**: Both functions approach negative infinity as \( x \) approaches 0, but they do not have a vertical asymptote since their domains start from \( x = 0 \) and go to \( x = \infty \).
3. **Horizontal Asymptote**: Neither function has a horizontal asymptote because they both increase without bound as \( x \) approaches infinity.
4. **Shape**: Both functions are increasing and have a similar curve shape typical of logarithmic functions.
5. **Rate of Increase**: The base of the logarithm affects the rate of increase. \( \log_3 x \) grows faster than \( \log_7 x \) for the same values of \( x \) because the larger base (7) results in a slower growth rate compared to a smaller base (3).
6. **X-Intercepts**: Both functions have an x-intercept at \( x = 1 \) because \( \log_b(1) = 0 \) for any base \( b > 0 \).
Considering these points, the most accurate statement that describes the similarities and differences is:
**C. Functions f(x) and g(x) will have similar shapes but will increase at different rates.**
This captures that both functions have a similar shape typical of logarithmic functions, but their growth rates differ due to their respective bases.