Question

In exponential function class $f\left(x\right)=b^x$ where $0<b<1$, greater rate of decrease in $x$ is dependant on

• A. smaller magnitude of base $y$
• B. smaller magnitude of base $x$
• C. smaller magnitude of base $a$
• D. smaller magnitude of base $b$

Answer 1

The correct option is D smaller magnitude of base $b$

$f\left(x\right)=b^x;0<b<1$

$\Rightarrow {\log }_{b}f\left(x\right)={\log }_b{b}^x$

$\Rightarrow x={\log }_{b}f\left(x\right)=\frac{\ln f\left(x\right)}{\ln \left(b\right)}$

$\Rightarrow \frac{dx}{dt}=\frac{1}{f\left(x\right)}.\frac{1}{\ln \left(b\right)}.\frac{df\left(x\right)}{dt}$

$\Rightarrow$ Rate of decrease is dependent on the smaller magnitude of base $b$.