Find $x\leq0$ that satisfies $ \ln\left(1+e^x\right) - \ln\left(1-e^x\right) \le \ln\left(1+4e^x\right)$

Question

Find $x \le 0$ that satisfies:

$$ \ln\left(1+e^x\right) - \ln\left(1-e^x\right) \le \ln\left(1+4e^x\right)$$

I have tried using the natural log laws to simplify the expressions but I always end up in a dead end. Would anyone just push me into the right direction with a slight hint?

Thanks in advance!

Answer 1

Let $e^x=t$.

Thus, $0<t<1$ and we need to solve $$\frac{1+t}{1-t}\leq1+4t$$ or $$4t^2-2t\leq0$$ or $$0\leq e^x\leq\frac{1}{2}$$ or $$x\leq-\ln2.$$