Prove that the function f(x)= _ e x is increasing on (0, ). - Vidyadip

Question

Prove that the function $\text{f}(\text{x})=\log_{\text{e}}\text{x}$ is increasing on $(0,\infty).$

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Answer

Let $\text{x}_1,\text{x}_2\in(0,\infty)$ such that $x_1 < x_2.$
Then $x_1 < x_2 $ Implies that_$\log_{\text{e}}\text{x}_1<\log_{\text{e}}\text{x}_2$
Implies that $f(x_1) < f(x_2)$
$\therefore x_1 < x_2$ Implies that $\text{f}(\text{x}_1)<\text{f}(\text{x}_2),\forall\ \text{x}_1,\text{x}_2\in(0,\infty)$
Therefore, $f(x)$ is increasing on $(0,\infty)$

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