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APPLICATION OF DERIVATIVES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF DERIVATIVES3 Marks
Question
Prove that the function $\text{f}(\text{x})=\log_{\text{e}}\text{x}$ is increasing on $(0,\infty)$ if $a > 1$ and decreasing on $(0,\infty)$ if $0 < a < 1.$
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Answer

Case I: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 $(0,\infty).$

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