Inequality for negative logarithms? Given 0 < x < y < 1 , is it pos

Madilyn Schroeder

Madilyn Schroeder

Answered question

2022-06-02

Inequality for negative logarithms?
Given 0 < x < y < 1, is it possible to prove the following result:
l n x l n y < 1
Thanks

Answer & Explanation

Cynthia Logan

Cynthia Logan

Beginner2022-06-03Added 2 answers

Take the inequality
ln x ln y < 1
And multiply by ln y:
ln x > ln y
(since y < 1, ln y < 0)
Now you can apply e x (which is monotone increasing) to both sides, to get
x > y
which contradicts the assumptions.
Baylee Newman

Baylee Newman

Beginner2022-06-04Added 3 answers

Since y < 1 log y < 0 , we get that>br? log x log y < 1 log x > log y
Yet we know that
( log x ) = 1 x > 0 log x is monotone increasing
and thus the inequality is wrong.

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