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I am self-studying for an actuarial exam and I encountered the following:

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The author seems to suggest that if we want to find the maximum of a function $f(x)$ with respect to $x$:

  1. We can drop any multiplicative constant.

  2. We can take the natural log of $f(x)$ and $\ln(f(x))$ has the same maximum as $f(x)$.

Property 1 is intuitive. If $f(x) = k\cdot g(x)$, then setting $f'(x) = k \cdot g'(x) = 0$ will result in $g'(x) = 0$.

I'm not necessarily convinced of property 2.

Why is it that $f(x)$ has the same maximum as $\ln(f(x))$? I took several semesters of calculus/analysis in university, and I don't recall this property.

Joseph DiNatale
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  • $\ln(x)$ is strictly increasing. That is, $x>y\iff \ln x>\ln y$ (trusting everything is defined, of course). Any strictly increasing function preserves max. – lulu Jul 26 '16 at 14:56
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    It doesn't have the same maximum, but it reaches a max at the same place, for $\ln$ is an increasing function. – André Nicolas Jul 26 '16 at 14:57
  • Re: "drop multiplicative constant" - This holds only for positive constants. A negative constant would swap the roles of maxima and minima – Hagen von Eitzen Jul 26 '16 at 15:12

1 Answers1

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Good question, this follows because the function $log$ is strictly increasing. This means that whenever $x > y$, $log(x)$ is greater than $log(y)$.

So in this case, if you can find the value x, such that $f(x)$ is maximised. Then that same value $x$, must be the maximum of $log(f(x))$.

I hope that makes sense (and apologies for the formatting, I'm figuring out how to do the mathematical format in the answer!),

Kevin

(note: the value of $log(f(x))$ is not equal to the value of $f(x)$, but I the question means that both are maximised for the same value of $x$)

Kevin C
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