A question on inequality equivalenceI'm doing a textbook question and I'm stuck in the very...

Mylee Underwood

Mylee Underwood

Answered

2022-07-09

A question on inequality equivalence
I'm doing a textbook question and I'm stuck in the very last end...
I got the following function on S:
f ( S ) = S q ( 1 q S ) p + q S c, where p := 1 q and 0 < p , q < 1.
Now the first forward difference of f(S), so Δ ( S ) := f ( S + 1 ) f ( S ) is Δ ( S ) = 1 q S ( p c + q ) and this function is increasing in S.
Now I need to find the value of S that satisfies Δ ( S ) 0.
We can then write an inequality:
q S 1 ( p c + q )
But now when I want to look for S, we can take logarithms and find:
S l n ( p c + q ) l n ( q )
But the textbook says that the equivalent inequality to q S 1 ( p c + q ) is: S l n ( p c + q ) l n ( q )
Why is this?
They also say in the question: "using the fact of ln ( q ) < 0.
Any help is appreciated!

Answer & Explanation

Kaylie Mcdonald

Kaylie Mcdonald

Expert

2022-07-10Added 19 answers

Step 1
The reason why the inequality sign is reversed is because q < 1, so log q ( x ) is actually a decreasing function.
Step 2
Therefore applying log to both sides will reverse the direction of inequality.
EnvivyEvoxys6

EnvivyEvoxys6

Expert

2022-07-11Added 7 answers

Explanation:
When you divide by a negative number the direction of the inequality get reversed. For example: 5 < 10 but when dividing by -1 the inequality we have to change the direction: 5 > 10.

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