Increasing and Decreasing Functions — Maths STD 12 Science — Question

When $\text{a}>1$

Let $\text{x}_1,\text{x}_2\in(0,\infty)$

We have

$\text{x}_1<\text{x}_2$

$\Rightarrow\log_{\text{a}}\text{x}_1<\log_{\text{a}}\text{x}_2$

$\Rightarrow\text{f}(\text{x}_1)<\text{f}(\text{x}_2)$

Thus, f(x) is increasing on $(0,\infty)$

Case II:

When $0<\text{a}<1$

$\text{f}(\text{x})=\log_{\text{a}}\text{x}=\frac{\log\text{x}}{\log\text{a}}$

When $\text{a}<1\Rightarrow\log_\text{a}<0$

Let $\text{x}_1<\text{x}_2$

_{$\Rightarrow\log\text{x}_1<\log\text{x}_2$}

_{$\Rightarrow\frac{\log\text{x}_1}{\log_\text{a}}>\frac{\log\text{x}_2}{\log_\text{a}}$ $[\because\ \log\text{a}<0]$}

_{$\Rightarrow\text{f}(\text{x}_1)>\text{f}(\text{x}_2)$}

So, f(x) is increasing on $(0,\infty).$