The correct statement that describes the similarities and differences between the logarithmic functions \( f(x) = \log_3 x \) and \( g(x) = \log_7 x \) is:
"Functions \( f(x) \) and \( g(x) \) will have similar shapes but will increase at different rates."
Explanation:
- Both functions are logarithmic functions and share the same general shape (they both increase slowly and approach infinity as \( x \) increases).
- However, they have different bases (3 and 7), which means they will increase at different rates. Specifically, \( g(x) = \log_7 x \) will increase more slowly than \( f(x) = \log_3 x \) because the base of \( g(x) \) is larger.
- Both functions have an x-intercept at \( x = 1 \) (since \( \log_b(1) = 0 \) for any base \( b \)), meaning they do not differ in their x-intercepts.
- Neither function has horizontal or vertical asymptotes; they both approach infinity and have a vertical asymptote at \( x = 0 \).
Thus, the key differentiation lies in their rates of growth rather than their shapes, intercepts, or asymptotes.
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