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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.

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  • $\begingroup$ Am I to assume $x>0$? $\endgroup$ – Fimpellizieri Nov 17 '17 at 4:22
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Hint: Logarithms have the following property:

$$\frac{\log_ax}{\log_ay}=\log_yx$$

Super Hint: Show that

$$f(x) \text{ is monotone} \iff \exp(f(x)) \text{ is monotone}$$

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  • $\begingroup$ Yes, thank you. Since y is a function of x rather than a constant, the monotonicity is still difficult to analyze. $\endgroup$ – zhang haiyang Nov 8 '17 at 4:53
  • $\begingroup$ See my super hint. $\endgroup$ – Fimpellizieri Nov 8 '17 at 5:05
  • $\begingroup$ 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. $\endgroup$ – zhang haiyang Nov 8 '17 at 8:06

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