# Prove the monotonicity of $q\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$

Here has a function $q(x)=\frac{f(x)}{g(x)}$, where

$$f(x)=\log_2\left(1+\frac{1}{a_1 x+b_1}\right),$$ $$g(x)=\log_2\left(1+\frac{1}{a_2 x+b_2}\right)$$ $$a_1,a_2,b_1,b_2 >0.$$

How to prove the monotonicity of $q(x)$? Since the first derivative is very complicated, it is difficult to analyze.

• Am I to assume $x>0$? – Fimpellizieri Nov 17 '17 at 4:22

$$\frac{\log_ax}{\log_ay}=\log_yx$$
$$f(x) \text{ is monotone} \iff \exp(f(x)) \text{ is monotone}$$
• Thanks for your helpful hint. The monotonicity of $q\left(x\right)$ depends on the relationship among a1, a2, b1, and b2. I want to reveal the impact of these constants on monotonicity, but it seems hard because the first derivative is so complicated. – zhang haiyang Nov 8 '17 at 8:06