Question:
Mark the correct alternative in the following:
Function $f(x)=\log _{a} x$ is increasing on $R$, if
A. $0 B. $a>1$ C. $a<1$ D. $a>0$
Solution:
Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)>0$ for all $x \in(a, b)$
$f(x)=\log _{a} x$
$\frac{\mathrm{d}(\mathrm{f}(\mathrm{x}))}{\mathrm{dx}}=\frac{1}{\mathrm{x} \log _{\mathrm{e}} \mathrm{a}}=\mathrm{f}^{\prime}(\mathrm{x})$
For increasing $f^{\prime}(x)>0$
$\Rightarrow \frac{1}{x \log _{e} a}>0$
For $\log a>1$