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#### please solve rd sharma class 12 chapter 16 Increasing and Decreasing Functions exercise 16.1 , question 2 maths textbook solution

$f(x)$  is decreasing on$(0,\infty )$if $0

Hint:

1. $f(x)$is increasing on$(a,b)$, if the values of $f(x)$ increase with the increase in the values of x.
2. $f(x)$is decreasing on$(a,b)$, if the values of $f(x)$ decrease with the increase in the values of x.

Given:

Here given that,

$f(x)=\log _{a} x$

To prove:

Function $f(x)=\log _{a} x$ is increasing on $(0,\infty )$, if $a>1$ and decreasing on $(0,\infty )$ , if $0

Solution:

Here we have two cases:

Case I: when $a>1$

Let us consider $x_{1},x_{2}\epsilon (0,\infty )$

Such that $x_{1}
$\begin{array}{ll} \Rightarrow & \log _{a} x_{1}<\log _{a} x_{2} \\ \Rightarrow & f\left(x_{1}\right)

Thus, $f(x)$ is increasing in$(0,\infty )$if $a>1$

Case II: when $0

We have $f(x)=\log _{a} x$

We know that $\log _{a} x=\frac{\log x}{\log a}$

Therefore, $f(x)=\frac{\log x}{\log a}$

When $a<1$, then $a<0$

Let $x_{1}

Taking log on both sides,

\begin{aligned} &\Rightarrow \quad & \log x_{1}<\log x_{2} \\ &\Rightarrow & \frac{\log x_{1}}{\log a}>\frac{\log x_{2}}{\log a} \\ &\Rightarrow & f\left(x_{1}\right)>f\left(x_{2}\right) \end{aligned}

Thus, $f(x)$ is decreasing on $\left ( 0,\infty \right )$ if $0