# Please help solve this, I can't come up with a solution. The question is to find f? Given #f:(0,+oo)->RR# with #f(x/e)<=lnx<=f(x)-1 , x in (0,+oo)#

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The answer is f(x) = lnx +1 but how do i prove it?

The answer is f(x) = lnx +1 but how do i prove it?

##### 2 Answers

#### Explanation:

We split the inequality into 2 parts:

Let's look at (1):

We rearrange to get

Let's look at (2):

We assume

From the 2 results,

Assume a form then use the bounds.

#### Explanation:

Based on the fact that we see that f(x) bounds ln(x), we might assume that the function is a form of ln(x). Let's assume a general form:

Plugging in the conditions, this means

We can subtract

Flipping,

If we want this to be true for all x, we see that the upper bound is a constant and

So we have only the solution with