Question

Function f(x) = log_a x is increasing on R, if
(a) 0 < a < 1
(b) a > 1
(c) a < 1
(d) a > 0

Answer 1

(b) a > 1

$$\begin{gathered} f\left(x\right)={\log }_{a}x=\frac{\log x}{\log a} \\ f\text{'}\left(x\right)=\frac{1}{x\log a} \\ \text{Given:}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) is increasing on\hspace{0.17em}}\mathit{\text{R.}} \\ \Rightarrow f\text{'}\left(x\right)>0,\forall x\in R \\ \Rightarrow \frac{1}{x\log a}>0,\forall x\in R \\ \Rightarrow a>1 \end{gathered}$$