The correct statement that describes the similarities and differences between the 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:
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Similar Shapes: Both logarithmic functions are concave and increase; they have the same general shape characteristic of logarithmic functions.
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Different Rates of Increase: The base of the logarithm affects the rate at which the function increases. Since \( \log_3 x \) has a smaller base than \( \log_7 x \), \( f(x) \) increases faster than \( g(x) \).
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Horizontal Asymptotes: Both functions approach negative infinity as \( x \) approaches 0 from the right, meaning they do not have different horizontal asymptotes.
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X-intercepts: Both functions have the same x-intercept, which is \( x = 1 \), since \( \log_b(1) = 0 \) for any base \( b \).
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Vertical Asymptotes: Neither function has vertical asymptotes.
Thus, the most accurate summary of the differences between \( f(x) \) and \( g(x) \) is their rate of increase, confirming that they will have similar shapes but grow at different rates.