Substituting one inequality in another inequality

Question

Suppose I have two inequalities:

$$f(n) \leqslant a × g(n)$$ and

$$2^{f(n)} \leqslant b × 2^{g(n)}$$

How can I substitute the equation 1 in equation 2?

Thanks

Answer 1

You cannot, if you want to make sure you obtain a true inequality. Your second inequality is equivalent to $$f(n)\leq \log_2 b+g(n)$$ and this could be stronger than your first inequality, meaning $$f(n)\leq \log_2b+g(n)\leq a\cdot g(n).$$