When compared to f(x) = ln(x), which of the following will increase faster? Please justify your answer. A.)  f(x) = 0.5 ln(x) B.)  f(x) = 5 ln(x) C.)  f(x) = ln(5x)

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The rate of increase of a function f(x) is the derivative of the function f'(x).

For f(x) = ln(x), `f'(x) = 1/x`

For f(x) = 0.5*ln(x), `f'(x) = 0.5/x`

For f(x) = 5*ln(x), `f'(x) = 5/x`

For f(x) = ln(5x) = ln 5 + ln(x), `f'(x) = 1/(x)`

From the derivatives of the functions it is seen that only `5/x` is greater than `1/x` .

The function f(x) = 5*ln(x) or option B increases at a rate faster than f(x) = ln(x).

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