Proving inequalities with logarithm

Think about the properties of integrals.In summary, the original problem can be solved by first determining if the function ( (n+1)/n )^(n+1) is monotonically increasing or decreasing for positive n. Then, find the asymptote of this function as n approaches infinity using the definition of e. Alternatively, the initial inequality can be shown directly using a clever integral, by replacing (log(n+1)-log(n)) with an integral and manipulating it to obtain the desired inequality.
  • #1
japplepie
93
0
I need to prove:

(n+1)*(log(n+1)-log(n) > 1 for all n > 0.

I have tried exponentiating it and I got

( (n+1)/n )^(n+1) < e.

And from there I couldn't go any farther, but I do know that it is true by just looking at its graph.

Could anybody help me please?
 
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  • #2
Correction:
japplepie said:
( (n+1)/n )^(n+1) > e.
As for the solution itself, first find out whether the function
$$
f(n) = \left( \frac{n+1}{n} \right)^{(n+1)}
$$
is a monotonically decreasing or increasing function for positive ##n## (it should be either of them). Then find the asymptote of this function as ##n \rightarrow \infty## in terms of ##e## by making use of the definition of ##e##
$$
e = \lim_{n \to \infty} \left( 1+\frac{1}{n} \right)^n .
$$
 
  • #3
blue_leaf77 said:
Correction:

As for the solution itself, first find out whether the function
$$
f(n) = \left( \frac{n+1}{n} \right)^{(n+1)}
$$
is a monotonically decreasing or increasing function for positive ##n## (it should be either of them). Then find the asymptote of this function as ##n \rightarrow \infty## in terms of ##e## by making use of the definition of ##e##
$$
e = \lim_{n \to \infty} \left( 1+\frac{1}{n} \right)^n .
$$

I tried that, but to know if it is indeed monotonic I have to show that the selected expression in the term the picture I attached is always < 1 or > 1.

So to proof of the original problem requires a proof for:
n*(log(n+1)-log(n)) < 1

Right now it looks pretty circular.
 

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  • #4
japplepie said:
I tried that, but to know if it is indeed monotonic I have to show that the selected expression in the term the picture I attached is always < 1 or > 1.
You don't need that, there is a more direct way.

You can also show the initial inequality directly with a clever integral. Can you rewrite ln(n+1)-ln(n) somehow?
 
  • #5
mfb said:
You don't need that, there is a more direct way.

You can also show the initial inequality directly with a clever integral. Can you rewrite ln(n+1)-ln(n) somehow?
mfb said:
You don't need that, there is a more direct way.

You can also show the initial inequality directly with a clever integral. Can you rewrite ln(n+1)-ln(n) somehow?

Can I have more clues please? I'm getting nowhere plus I'm not that good with integral calculus since we were never taught this in my high school & university.
 
  • #6
What is ##\int_a^b \frac{dx}{x}##?
 
  • #7
mfb said:
What is ##\int_a^b \frac{dx}{x}##?
log(x), but I still really can't where I'm supposed to be headed.
 
  • #8
It is not log(x). The definite integral not depend on x at all, but only on a and b.
 
  • #9
mfb said:
It is not log(x). The definite integral not depend on x at all, but only on a and b.
log(b)-log(a) ?
 
  • #10
Right, the difference between two logarithms - that's what you have in your original inequality.
 
  • #11
mfb said:
Right, the difference between two logarithms - that's what you have in your original inequality.
I think you're implying that I integrate both sides, but how do I choose the value of a and b?
 
  • #12
No, don't integrate. Replace (log(n+1)-log(n)) with an integral, and then find some way to get the inequality you need.
 

1. How can logarithms be used to prove inequalities?

Logarithms can be used to prove inequalities by taking the logarithm of both sides of the inequality and using logarithm rules to simplify the expression. This allows for a comparison of the values on each side of the inequality.

2. What are the properties of logarithms that can be used to prove inequalities?

The properties of logarithms that can be used to prove inequalities include the power rule, product rule, quotient rule, and change of base rule. These rules allow for the manipulation of logarithmic expressions in order to compare them to each other.

3. Can logarithms only be used with numerical inequalities?

No, logarithms can also be used to prove inequalities with variables. In these cases, the properties of logarithms can be used to solve for the variable and determine the range of values that satisfy the inequality.

4. Are there any limitations to using logarithms to prove inequalities?

One limitation is that logarithms can only be used to prove strict inequalities (>, <) and not non-strict inequalities (≥, ≤). Additionally, logarithms cannot be used to prove inequalities with negative numbers or zero as the base.

5. How do I know if I have correctly proven an inequality using logarithms?

If you have correctly proven an inequality using logarithms, the resulting expression should clearly show which side of the inequality is greater. Additionally, you can check your work by plugging in specific values for the variables and comparing the results on each side of the inequality.

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