Question

Prove that the function f(x) = log_e x is increasing on (0, ∞).

Answer 1

$$\begin{gathered} \text{Let}{x}_1,x_2\in \left(0,\infty \right)\text{such that}{x}_1<x_2.\text{Then,} \\ {x}_1<x_2 \\ \Rightarrow {\log }_e{x}_1<{\log }_e{x}_2 \\ \Rightarrow f\left(x_1\right)<f\left(x_2\right) \\ \text{∴}{x}_1<x_2\Rightarrow f\left(x_1\right)<f\left(x_2\right),\forall x_1,x_2\in \left(0,\infty \right) \\ \text{So,}f\left(x\right)\text{is increasing on}\left(0,\infty \right)\text{.} \end{gathered}$$