Question
Compare the two logarithmic functions f(x) = logg x and g (x) = log, x. Which statement correctly describes the
similarities and differences between these two functions? 1 point)
A. • Functions f(x) and g(x) will have similar shapes but will increase at different rates.
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 have different x-intercepts.
D. • Functions f(x) and g(x) will have similar shapes but will have different vertical asymptotes.
similarities and differences between these two functions? 1 point)
A. • Functions f(x) and g(x) will have similar shapes but will increase at different rates.
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 have different x-intercepts.
D. • Functions f(x) and g(x) will have similar shapes but will have different vertical asymptotes.
Answers
GPT-4o mini
To compare the two logarithmic functions \( f(x) = \log_g x \) and \( g(x) = \log x \), we need to analyze their properties.
1. **Base of the Logarithm**:
- \( f(x) \) uses a base \( g \) (which can be any positive number except 1).
- \( g(x) \) generally implies the base 10 logarithm if not specified, or the natural logarithm if we use \( \ln \).
2. **Shape**:
- Both functions will have a similar general shape: they both will be increasing functions that approach \(-\infty\) as \( x \) approaches 0 from the right.
3. **Asymptotes**:
- Both functions have a vertical asymptote at \( x = 0 \) (they approach \(-\infty\) as \( x \to 0^+\)), but they do not have horizontal asymptotes because they will both increase without bound as \( x \) increases.
4. **X-Intercepts**:
- Both functions have an x-intercept at \( x = 1 \) because \( \log_b(1) = 0 \) for any base \( b > 0 \).
Given these points, the correct statement that describes both functions is:
**A. Functions f(x) and g(x) will have similar shapes but will increase at different rates.**
The rates of increase depend on the bases of the logarithms; a logarithm with a larger base increases more slowly than a logarithm with a smaller base. If \( g < 1 \), \( f(x) \) will increase faster than \( g(x) \) (if \( g(x) = \log_{10}x \)), and vice versa.
1. **Base of the Logarithm**:
- \( f(x) \) uses a base \( g \) (which can be any positive number except 1).
- \( g(x) \) generally implies the base 10 logarithm if not specified, or the natural logarithm if we use \( \ln \).
2. **Shape**:
- Both functions will have a similar general shape: they both will be increasing functions that approach \(-\infty\) as \( x \) approaches 0 from the right.
3. **Asymptotes**:
- Both functions have a vertical asymptote at \( x = 0 \) (they approach \(-\infty\) as \( x \to 0^+\)), but they do not have horizontal asymptotes because they will both increase without bound as \( x \) increases.
4. **X-Intercepts**:
- Both functions have an x-intercept at \( x = 1 \) because \( \log_b(1) = 0 \) for any base \( b > 0 \).
Given these points, the correct statement that describes both functions is:
**A. Functions f(x) and g(x) will have similar shapes but will increase at different rates.**
The rates of increase depend on the bases of the logarithms; a logarithm with a larger base increases more slowly than a logarithm with a smaller base. If \( g < 1 \), \( f(x) \) will increase faster than \( g(x) \) (if \( g(x) = \log_{10}x \)), and vice versa.