A question on inequality equivalenceI'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 &lt; p , q &lt; 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 ) &lt; 0.Any help is appreciated!

Question

A question on inequality equivalenceI&#039;m doing a textbook question and I&#039;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  &amp;lt;  p  ,  q  &amp;lt;  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: &quot;using the fact of   ln  ⁡  (  q  )  &amp;lt;  0.Any help is appreciated!

Kaylie Mcdonald · Accepted Answer

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

EnvivyEvoxys6 · Accepted Answer

Explanation:When you divide by a negative number the direction of the inequality get reversed. For example:   5  &amp;lt;  10  but when dividing by -1 the inequality we have to change the direction:   −  5  &amp;gt;  −  10.