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 $(0,\infty).$

✓

Answer

Let $x _1, x _2 \in(0, \infty)$ such that $x _1< x _2$.
Then $x _1< x _2$ Implies that $\log _{ e } x _1<\log _{ e } x _2$ Implies that $f \left( x _1\right)< f \left( x _2\right)$
$\therefore x _1< x _2$ Implies that $f \left( x _1\right)< f \left( x _2\right), \forall x _1, x _2 \in(0, \infty)$
Therefore, $f(x)$ is increasing on $(0, \infty)$

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