# is it possible to solve this inequality for L?

Hello. Let $$C_1,C_2,C_3,q_1,q_2, \epsilon \geq 0$$ and $$h \in (0,1)$$ it is possible to solve this inequality for $$L \in \mathbb{N}$$?:
$$C_1h^{q_1L}+2C_2C_3h^{q3L}\leq \epsilon$$

Ive solved it for some special cases of $$q_1$$ and $$q_2$$ and im starting to wonder if its even possible to solve this straight forward.

• You can "unclutter" the expression as $ab^{c^L}+de^{f^L}\le1$ I guess. I doubt there is a closed-form solution. – Yves Daoust Dec 3 '18 at 17:10
• I think so too at least i have solved it in the cases $q_1=2q_3$ and vice versa. For the other cases I will probably solve it numerically. I just thought maybe some1 has an idea that i didnt have – StefanWK Dec 3 '18 at 17:23
• Mh, I am not sure that my retranscript is correct. Isn't it of the simpler form $ab^L+cd^L\le1$ ? – Yves Daoust Dec 3 '18 at 17:28

The inequation can be rewritten

$$ab^L+cd^L\le1$$ with $$b,d<1$$.

This is a decreasing function of $$L$$. As $$L\in N$$, a simple linear search by increasing values of $$L$$ can do. For better efficiency, you can use an exponential search (doublings) followed by a dichotomic one.