# Are there any mechanisms to solve a log quadratic inequality?

I have an inequality as follows

$$y \log y - (1-y) \log(1-y) < k$$

where $k$ is a constant. I tried solving the equation using exponential on both sides, but it does not come to a proper form where I can get the solution. How can I proceed to solve this inequation?

• Such equations with the unknown outside and inside a transcendental function usually have no analytic solution. You need to resort to numerical methods, after solving the root separation issue. – Yves Daoust Mar 29 '16 at 9:34
• I do not see an equation, but an inequality. – Julián Aguirre Mar 29 '16 at 9:48

The given function is defined for $x\in[0,1]$ and takes values in $[-\ln(2),0]$. It reminds the shape of a parabola and is symmetric about $x=\dfrac12$.
Hence, for any $k\in(-\ln(2),0)$, you can start numerical resolution of the equation, for instance by regula falsi with $x_0=0,x_1=\dfrac12$.
After convergence to $\hat x$, the solution of the inequation is the open interval $(\hat x,1-\hat x)$.