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I have an expression involving very small probabilities $A,B,C,D$.

$$x \ge \frac{k_1B}{A + k_2B}$$ $$x \le \frac{k_1C}{D + k_2C}$$

Is there any way for me to check if $x$ satisfies the above inequalities using $\log(A),\log(B),\log(C)$ and $\log(D)$?

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You could try inverting the inequalities as follows (example, assuming $k_1$ is positive):$$\frac 1x \leq \frac A {k_1B}+\frac {k_2}{k_1} \text{ or } \frac {k_1}x - k_2 \leq \frac A {B}$$

Then taking logarithms of each side (providing LHS is positive) isolates $\log (A)$ and $\log(B)$. Not sure whether that is what you want though.

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