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 $\text{x}_1,\text{x}_2\in(0,\infty)$ such that x₁ < x₂. Then

x₁ < x₂

Implies that$\log_{\text{e}}\text{x}_1<\log_{\text{e}}\text{x}_2$

Implies that f(x₁) < f(x₂)

$\therefore$ x₁ < x₂ 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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