I'm trying to solve this inequality for N where c, p, a are constants. For the 3rd equation, I flipped the inequality because I multiplied both sides by -1. But I'm not sure if the inequality changes when taking the log of both sides?

$\displaystyle 10\log \left(\frac{1}{1+\left(\frac{c}{p}\right)^{2N}}\ri ght)>-a$

$\displaystyle -10\log \left(1+\left(\frac{c}{p}\right)^{2N}\right)>-a$

$\displaystyle 10\log \left(1+\left(\frac{c}{p}\right)^{2N}\right)<a$

$\displaystyle \log \left(1+\left(\frac{c}{p}\right)^{2N}\right)<\frac {a}{10}$

exponentiate both sides...

$\displaystyle \left(\frac{c}{p}\right)^{2N}<10^{a/10}-1$

take log of both sides...

$\displaystyle 2N \log \left(\frac{c}{p}\right)<\log \left(10^{a/10}-1\right)$

$\displaystyle N<\frac{\log \left(10^{a/10}-1\right)}{2 \log \left(\frac{c}{p}\right)}$