Why Does Maximizing a Function Also Maximize Its Logarithm?

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The discussion clarifies that maximizing a function f(x) also maximizes its logarithm ln(f(x)). This is established through the relationship between the derivatives of the function and its logarithm. Specifically, the derivative of ln(f) is f'/f, which is zero only when the numerator f' is zero, indicating critical points. Since ln(x) is an increasing function, the maximum of f(x) directly correlates with the maximum of ln(f(x)).

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the value same which maximizes the logarithm of the function and the plain form of the function
why??
please help me,
thanks
 
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Could you rephrase your question? It is hard to understand.
 
Do you mean, you have some function, f, and its logarithm, ln(f), and you want to find value of x that minimizes both f and ln(f)?

Assuming there are no "boundaries" then we are looking for critical points, where the derivative is 0 or does not exist. The derivative of f is f' and the derivative of ln(f) is f'/f. A fraction is 0 if and only if its numerator is 0. Assuming that f(x) is not 0, in which case ln(f(x)) would not exist, f'(x)/f(x) is 0 if and only if f'(x) is 0. Further, since ln(x) is an increasing function, a maximum for f(x) implies a maximum for ln(f(x)) and vice-versa.l
 

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